%------------------------------------------------------------------------------
% File     : ITP275_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00039_002338
% 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_00039_002338 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12812 (4211 unt;2056 typ;   0 def)
%            Number of atoms       : 26400 (10006 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17351 (2670   ~; 571   |;1406   &)
%                                         (2906 <=>;9798  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of FOOLs       : 1726 ( 697 fml;1029 var)
%            Number of X terms     :  978 (   0  []; 799 ite; 179 let)
%            Number of types       :  388 ( 387 usr)
%            Number of type conns  : 1893 (1268   >; 625   *;   0   +;   0  <<)
%            Number of predicates  :   83 (  80 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1611 (1611 usr; 423 con; 0-8 aty)
%            Number of variables   : 26013 (25260   !; 753   ?;26013   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 15:20:12.468
%------------------------------------------------------------------------------
% Could-be-implicit typings (387)
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    fun_nat_complex: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
    fun_int_complex: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Rat__Orat_J,type,
    fun_complex_rat: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Num__Onum_J,type,
    fun_complex_num: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
    fun_complex_nat: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
    fun_complex_int: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
    fun_set_rat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Num__Onum_J_M_Eo_J,type,
    fun_set_num_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    fun_set_nat_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
    fun_set_int_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    fun_Code_integer_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_nat_fun_o_o: $tType ).

tff(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    set_Extended_enat: $tType ).

tff(ty_n_t__fun_It__String__Ochar_Mt__Nat__Onat_J,type,
    fun_char_nat: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    fun_real_real: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Ochar_J,type,
    fun_nat_char: $tType ).

tff(ty_n_t__fun_It__Extended____Nat__Oenat_M_Eo_J,type,
    fun_Extended_enat_o: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
    fun_real_rat: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Num__Onum_J,type,
    fun_real_num: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
    fun_real_nat: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
    fun_real_int: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Real__Oreal_J,type,
    fun_rat_real: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Real__Oreal_J,type,
    fun_num_real: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    fun_nat_real: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Real__Oreal_J,type,
    fun_int_real: $tType ).

tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
    list_complex: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
    set_list_o: $tType ).

tff(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    fun_rat_rat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Num__Onum_J,type,
    fun_rat_num: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Nat__Onat_J,type,
    fun_rat_nat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Int__Oint_J,type,
    fun_rat_int: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Rat__Orat_J,type,
    fun_num_rat: $tType ).

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__List__Olist_I_Eo_J_M_Eo_J,type,
    fun_list_o_o: $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__fun_It__Complex__Ocomplex_M_Eo_J,type,
    fun_complex_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Complex__Ocomplex_J,type,
    fun_o_complex: $tType ).

tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
    fun_set_o_o: $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__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__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__fun_It__Real__Oreal_M_Eo_J,type,
    fun_real_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Real__Oreal_J,type,
    fun_o_real: $tType ).

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

tff(ty_n_t__fun_It__Rat__Orat_M_Eo_J,type,
    fun_rat_o: $tType ).

tff(ty_n_t__fun_It__Num__Onum_M_Eo_J,type,
    fun_num_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
    fun_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
    fun_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Rat__Orat_J,type,
    fun_o_rat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Num__Onum_J,type,
    fun_o_num: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
    fun_o_nat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
    fun_o_int: $tType ).

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

tff(ty_n_t__List__Olist_I_Eo_J,type,
    list_o: $tType ).

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

tff(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
    fun_o_o: $tType ).

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

tff(ty_n_t__Real__Oreal,type,
    real: $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 (1669)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_complex > fun_complex_o ).

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: $o > fun_o_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_real_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_o_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_nat_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_int_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_nat > fun_fu2408719596384163971_nat_o ).

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

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: $o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_real_real ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_int_int ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_real_real ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_int_int ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_nat > fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_list_nat_o > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_set_nat_o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_nat_real ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: real > fun_fu2598743018846156927al_rat ).

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

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_real_o > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_int_o > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_complex_o > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_real_o > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_int_o > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_complex_o > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_o > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_real_o > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_int_o > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_complex_o > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_complex_rat > fun_fu4396998534562958809ex_rat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_real_nat > fun_fu2707946803925496145al_nat ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_complex_nat > fun_fu1553612835185860561ex_nat ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_int_int > fun_int_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_real_complex > fun_fu3532930458344041007omplex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_nat_complex > fun_fu4302481349826706223omplex ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_int_complex > fun_fu3558165918884697647omplex ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_complex_rat > fun_fu4396998534562958809ex_rat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_real_complex > fun_se2317959235522472630omplex ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_int_complex > fun_se7807923622652057142omplex ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_real_real > fun_se5601468920065002932l_real ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_int_real > fun_se7886327339988957492t_real ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_complex_real > fun_se7589301288977092664x_real ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_real_rat > fun_se2428050584324877792al_rat ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_int_rat > fun_se937733209267607904nt_rat ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_complex_rat > fun_se8588693959369716324ex_rat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_real_nat > fun_se8808036921802555352al_nat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_int_nat > fun_se7317719546745285464nt_nat ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_real_complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_int_complex > fun_int_o ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_real_rat > fun_real_o ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_real_o > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_int_o > fun_fu4328765270694171897t_real ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_complex_o > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_real_o > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_int_o > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_complex_o > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_real_o > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_int_o > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_complex_o > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_real_o > fun_fu5645595436833112697al_int ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_re6978948838062957830nteger > fun_Co1395685470531185632nteger ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_re6978948838062957830nteger > fun_Co4425031189514048897real_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_re6978948838062957830nteger > fun_Co4425031189514048897real_o ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_int_Code_integer > fun_Co3011819262099091808nteger ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_int_Code_integer > fun_Co8353275306710069249_int_o ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_int_Code_integer > fun_Co8353275306710069249_int_o ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_co8153928105989009288nteger > fun_Co8416587583108374242nteger ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_co8153928105989009288nteger > fun_Co8975588572093281791plex_o ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_co8153928105989009288nteger > fun_Co8975588572093281791plex_o ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_nat_Code_integer > fun_Co1375048237525739357_nat_o ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_Code_integer > fun_Co1375048237525739357_nat_o ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_real_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_real_nat > fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_real_nat > fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_int_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_int_nat > fun_nat_fun_int_o ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_int_nat > fun_nat_fun_int_o ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_complex_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_complex_nat > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_complex_nat > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_real_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_real_int > fun_int_fun_real_o ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_real_int > fun_int_fun_real_o ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_complex_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_complex_int > fun_in5065666251602127472plex_o ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_complex_int > fun_in5065666251602127472plex_o ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_int > fun_int_fun_nat_int ).

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

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

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

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: complex > fun_fu4692242605864086655ex_int ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: real > fun_fu5645595436833112697al_int ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: $o > fun_fu2803337334652888941o_real ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: $o > fun_fu9039538216874201369_o_rat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_rat > fun_fu8328862064278946247at_rat ).

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

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_nat_int > fun_nat_int ).

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_complex > fun_nat_complex ).

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

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

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_nat_real > fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_rat > fun_nat_rat ).

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

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

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

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: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_nat_real ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_real ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: nat > fun_na3244735409550470327at_nat ).

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

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_real_o ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: real > fun_int_o ).

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

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_o ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: set_nat > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: nat > fun_num_option_num ).

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

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: int > fun_in5570076352459579423_int_o ).

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

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_real_real ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_re4238438383274954274t_real ).

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

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

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

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_real_real ).

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

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_nat_real > fun_fu6119738925743952495t_real ).

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

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

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_real_real ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_nat_set_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: real > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_real_o ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_real_o ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

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

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

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: fun_num_fun_num_num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: ( nat * code_integer ) > code_integer ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: ( code_integer * code_integer ) > code_integer ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1345352211410354436nteger: ( nat * code_integer ) > code_integer ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
    bit_se2119862282449309892nteger: fun_nat_Code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
    bit_se1080825931792720795nteger: ( code_integer * code_integer ) > code_integer ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
    bit_se9216721137139052372nteger: code_integer > fun_nat_o ).

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_o ).

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

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
    bit_un1837492267222099188nd_num: fun_nu1049378303802735384on_num ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
    bit_un6178654185764691216or_num: fun_nu1049378303802735384on_num ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: fun_nu1049378303802735384on_num ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: fun_nu1049378303802735384on_num ).

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

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

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

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

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

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

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

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

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: fun_set_nat_nat ).

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

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: fun_set_nat_nat ).

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

tff(sy_c_Complex_OArg,type,
    arg: fun_complex_real ).

tff(sy_c_Complex_Ocis,type,
    cis: fun_real_complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: fun_complex_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > fun_real_complex ).

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: fun_complex_real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: fun_complex_real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: fun_complex_complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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

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

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

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

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

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

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod_int_int ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: ( num * num ) > product_prod_int_int ).

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

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

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: ( num * product_prod_int_int ) > product_prod_int_int ).

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

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
    semiri5044797733671781792omplex: fun_nat_complex ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: fun_nat_nat ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: fun_nat_real ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
    invers8013647133539491842omplex: fun_complex_complex ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: fun_rat_rat ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: fun_real_real ).

tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

tff(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

tff(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
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tff(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( fun_nat_nat * filter_nat * filter_nat ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
    filterlim_nat_real: ( fun_nat_real * filter_real * filter_nat ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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tff(sy_c_Finite__Set_Ocard_001_Eo,type,
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tff(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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tff(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
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tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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tff(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
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tff(sy_c_Finite__Set_Ofinite_001_Eo,type,
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tff(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
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tff(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
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tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
    finite_finite_list_o: set_list_o > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
    finite8712137658972009173omplex: set_list_complex > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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tff(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
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tff(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
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tff(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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tff(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
    finite1152437895449049373et_nat: fun_set_set_nat_o ).

tff(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
    finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).

tff(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    bij_be1856998921033663316omplex: ( fun_complex_complex * set_complex ) > fun_set_complex_o ).

tff(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    bij_betw_nat_complex: ( fun_nat_complex * set_nat ) > fun_set_complex_o ).

tff(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
    bij_betw_nat_nat: ( fun_nat_nat * set_nat ) > fun_set_nat_o ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_C7848664941760151970nteger: ( fun_Co1561183069634336733nteger * fun_Co8499238673658658103nteger * code_integer ) > fun_Pr5876161306168176259nteger ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( fun_nat_nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_nat_real_nat: ( fun_nat_real * fun_nat_nat ) > fun_nat_real ).

tff(sy_c_Fun_Ocomp_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_f6739083541385432444nteger: ( fun_fu4800041728187187127nteger * fun_Co1981900195452921617nteger ) > fun_Co1561183069634336733nteger ).

tff(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: fun_nat_nat > fun_set_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
    inj_on_nat_char: fun_nat_char > fun_set_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
    inj_on_real_real: fun_real_real > fun_set_real_o ).

tff(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
    strict1292158309912662752at_nat: fun_nat_nat > fun_set_nat_o ).

tff(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
    the_in5290026491893676941l_real: ( set_real * fun_real_real ) > fun_real_real ).

tff(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
    gcd_Gcd_int: set_int > int ).

tff(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
    gcd_Gcd_nat: fun_set_nat_nat ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod_int_int ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_GCD_Ogcd__class_Ogcd_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
    gcd_gcd_int: int > fun_int_int ).

tff(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
    gcd_gcd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
    abs_abs_Code_integer: code_integer > code_integer ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
    abs_abs_complex: fun_complex_complex ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: fun_int_int ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
    abs_abs_rat: fun_rat_rat ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
    abs_abs_real: fun_real_real ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
    minus_8373710615458151222nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
    minus_minus_complex: complex > fun_complex_complex ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
    minus_3235023915231533773d_enat: ( extended_enat * extended_enat ) > extended_enat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > fun_int_int ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: rat > fun_rat_rat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: fun_re7681423839302404990l_real ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
    minus_minus_set_o: ( set_o * set_o ) > set_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    minus_811609699411566653omplex: ( set_complex * set_complex ) > set_complex ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
    minus_minus_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    minus_7954133019191499631st_nat: ( set_list_nat * set_list_nat ) > set_list_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: ( set_nat * set_nat ) > set_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
    minus_minus_set_real: ( set_real * set_real ) > set_real ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: ( set_set_nat * set_set_nat ) > set_set_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    minus_8169481405280535082plex_o: ( fun_complex_o * fun_complex_o ) > fun_complex_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Int__Oint_M_Eo_J,type,
    minus_7151557278095657260_int_o: ( fun_int_o * fun_int_o ) > fun_int_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    minus_5761417318614613624_nat_o: ( fun_list_nat_o * fun_list_nat_o ) > fun_list_nat_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Nat__Onat_M_Eo_J,type,
    minus_8972599623528345224_nat_o: ( fun_nat_o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Real__Oreal_M_Eo_J,type,
    minus_404528632214381484real_o: ( fun_real_o * fun_real_o ) > fun_real_o ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    minus_6087510627544308050_nat_o: ( fun_set_nat_o * fun_set_nat_o ) > fun_set_nat_o ).

tff(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
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tff(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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tff(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

tff(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
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tff(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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tff(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
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tff(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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tff(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: fun_num_fun_num_num ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
    plus_plus_rat: rat > fun_rat_rat ).

tff(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
    plus_plus_real: fun_re7681423839302404990l_real ).

tff(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
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tff(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
    sgn_sgn_int: fun_int_int ).

tff(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
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tff(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
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tff(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
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tff(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > fun_int_int ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
    times_times_num: fun_num_fun_num_num ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
    times_times_rat: rat > fun_rat_rat ).

tff(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
    times_times_real: fun_re7681423839302404990l_real ).

tff(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
    uminus1351360451143612070nteger: fun_Co8499238673658658103nteger ).

tff(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
    uminus1482373934393186551omplex: fun_complex_complex ).

tff(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: fun_int_int ).

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tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( fun_nat_nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: list_o > set_o ).

tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
    set_complex2: list_complex > set_complex ).

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

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

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__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

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_001_Eo,type,
    list_update_o: ( list_o * nat * $o ) > list_o ).

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

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

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

tff(sy_c_List_Olist__update_001t__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__VEBT____Definitions__OVEBT,type,
    list_u1324408373059187874T_VEBT: ( list_VEBT_VEBT * nat * vEBT_VEBT ) > list_VEBT_VEBT ).

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

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

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

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

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

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    nth_Product_prod_o_o: ( list_P4002435161011370285od_o_o * nat ) > product_prod_o_o ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    nth_Pr1649062631805364268_o_int: ( list_P3795440434834930179_o_int * nat ) > product_prod_o_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    nth_Pr5826913651314560976_o_nat: ( list_P6285523579766656935_o_nat * nat ) > product_prod_o_nat ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr6777367263587873994T_VEBT: ( list_P7495141550334521929T_VEBT * nat ) > produc2504756804600209347T_VEBT ).

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    nth_Pr6456567536196504476um_num: ( list_P3744719386663036955um_num * nat ) > product_prod_num_num ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    nth_Pr4606735188037164562VEBT_o: ( list_P3126845725202233233VEBT_o * nat ) > produc334124729049499915VEBT_o ).

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,
    nth_Pr4953567300277697838T_VEBT: ( list_P7413028617227757229T_VEBT * nat ) > produc8243902056947475879T_VEBT ).

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

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

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

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

tff(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
    product_o_int: ( list_o * list_int ) > list_P3795440434834930179_o_int ).

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

tff(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    product_o_VEBT_VEBT: ( list_o * list_VEBT_VEBT ) > list_P7495141550334521929T_VEBT ).

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

tff(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: ( list_nat * list_o ) > list_P7333126701944960589_nat_o ).

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__Num__Onum_001t__Num__Onum,type,
    product_num_num: ( list_num * list_num ) > list_P3744719386663036955um_num ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    product_VEBT_VEBT_o: ( list_VEBT_VEBT * list_o ) > list_P3126845725202233233VEBT_o ).

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

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

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

tff(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

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

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

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

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

tff(sy_c_List_Oreplicate_001t__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_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_o > fun_list_int_o ).

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

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

tff(sy_c_List_Ounion_001t__Int__Oint,type,
    union_int: ( list_int * list_int ) > list_int ).

tff(sy_c_List_Ounion_001t__Nat__Onat,type,
    union_nat: ( list_nat * list_nat ) > list_nat ).

tff(sy_c_List_Ounion_001t__VEBT____Definitions__OVEBT,type,
    union_VEBT_VEBT: ( list_VEBT_VEBT * 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_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

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

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

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Complex__Ocomplex,type,
    semiri3842193898606819883omplex: set_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: fun_na8797796813480665199d_enat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

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

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

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

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > 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_M_Eo_J_J,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > 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__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__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: fun_VEBT_VEBT_nat ).

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

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

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

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

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

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
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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__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_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__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

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

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o * $o ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

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

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

tff(sy_c_Topological__Spaces_Omonoseq_001t__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_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

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

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__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_Pr8877776288780692068_nat_o ).

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_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

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

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

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

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

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

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

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option_nat ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( fun_Pr9018117161326028078_int_o * product_prod_int_int ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( fun_Pr2028404973800837934_nat_o * product_prod_nat_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr8877776288780692068_nat_o * produc9072475918466114483BT_nat ) > $o ).

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

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

tff(sy_c_aa_001_Eo_001t__Complex__Ocomplex,type,
    aa_o_complex: ( fun_o_complex * $o ) > complex ).

tff(sy_c_aa_001_Eo_001t__Int__Oint,type,
    aa_o_int: ( fun_o_int * $o ) > int ).

tff(sy_c_aa_001_Eo_001t__Nat__Onat,type,
    aa_o_nat: ( fun_o_nat * $o ) > nat ).

tff(sy_c_aa_001_Eo_001t__Num__Onum,type,
    aa_o_num: ( fun_o_num * $o ) > num ).

tff(sy_c_aa_001_Eo_001t__Rat__Orat,type,
    aa_o_rat: ( fun_o_rat * $o ) > rat ).

tff(sy_c_aa_001_Eo_001t__Real__Oreal,type,
    aa_o_real: ( fun_o_real * $o ) > real ).

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

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

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

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

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

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod326357343816142457eger_o: ( fun_Co2081214437782444048eger_o * code_integer ) > fun_Code_integer_o ).

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__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__Num__Onum_J,type,
    aa_Cod6128727250572145423er_num: ( fun_Co8521658056209057742er_num * code_integer ) > fun_Code_integer_num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    aa_Cod1096751537565085541eger_o: ( fun_Co3544761417380347684eger_o * code_integer ) > fun_Co1766037875893669450eger_o ).

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__Complex__Ocomplex_M_Eo_J,type,
    aa_Cod7908732550325689320plex_o: ( fun_Co8975588572093281791plex_o * code_integer ) > fun_complex_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Complex__Ocomplex_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod1660996421839551523nteger: ( fun_Co8416587583108374242nteger * code_integer ) > fun_co8153928105989009288nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_Cod8477617969808203498_int_o: ( fun_Co8353275306710069249_int_o * code_integer ) > fun_int_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod1949805079358391713nteger: ( fun_Co3011819262099091808nteger * code_integer ) > fun_int_Code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_Cod1075288278386115654_nat_o: ( fun_Co1375048237525739357_nat_o * code_integer ) > fun_nat_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J_001t__fun_It__fun_It__Real__Oreal_Mt__Nat__Onat_J_Mt__fun_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    aa_fun1500080954902701606al_nat: ( fun_fu3768230810189712741al_nat * fun_real_nat ) > fun_fu3214020956574444655al_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun3209485225100452678real_o: ( fun_fu6510078244439976541real_o * fun_real_rat ) > fun_real_o ).

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

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__fun_It__Real__Oreal_Mt__Rat__Orat_J_Mt__fun_It__Real__Oreal_M_Eo_J_J,type,
    aa_fun8459995508950063326real_o: ( fun_fu5587483295140630005real_o * fun_real_rat ) > fun_fu6510078244439976541real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__fun_It__Real__Oreal_Mt__Rat__Orat_J_Mt__fun_It__Real__Oreal_Mt__Rat__Orat_J_J,type,
    aa_fun1537150169341160510al_rat: ( fun_fu7187499828406537085al_rat * fun_real_rat ) > fun_fu2598743018846156927al_rat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun2895569901277098738real_o: ( fun_fu3022329567491847689real_o * fun_real_real ) > fun_real_o ).

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

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_M_Eo_J_J,type,
    aa_fun3693225987673163702real_o: ( fun_fu5533088652784741581real_o * fun_real_real ) > fun_fu3022329567491847689real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    aa_fun5414376518349398074l_real: ( fun_fu9132278742410140537l_real * fun_real_real ) > fun_fu3077872570260440615l_real ).

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_fun751667557118977660_nat_o: ( fun_fu2408719596384163971_nat_o * fun_set_nat_o ) > fun_set_nat_o ).

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

tff(sy_c_fequal_001_Eo,type,
    fequal_o: $o > fun_o_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__List__Olist_It__Nat__Onat_J,type,
    fequal_list_nat: list_nat > fun_list_nat_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_real_fun_real_o ).

tff(sy_c_fequal_001t__Set__Oset_It__Nat__Onat_J,type,
    fequal_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_member_001_Eo,type,
    member_o: $o > fun_set_o_o ).

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

tff(sy_c_member_001t__Extended____Nat__Oenat,type,
    member_Extended_enat: extended_enat > fun_se8123728892507317397enat_o ).

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

tff(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: ( list_o * set_list_o ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: ( list_int * set_list_int ) > $o ).

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

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

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

tff(sy_c_member_001t__Num__Onum,type,
    member_num: num > fun_set_num_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > fun_set_rat_o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: real > fun_set_real_o ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > fun_set_set_nat_o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: ( vEBT_VEBT * set_VEBT_VEBT ) > $o ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_i____,type,
    i: nat ).

tff(sy_v_m____,type,
    m: 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 ).

tff(sy_v_x____,type,
    x: nat ).

% Relevant facts (10741)
tff(fact_0_ac,axiom,
    size_s6755466524823107622T_VEBT(treeList) = size_s6755466524823107622T_VEBT(treeList2) ).

% ac
tff(fact_1__C2_Ohyps_C_I3_J,axiom,
    m = na ).

% "2.hyps"(3)
tff(fact_2__C2_Ohyps_C_I2_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) ).

% "2.hyps"(2)
tff(fact_3_less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% less_exp
tff(fact_4_semiring__norm_I85_J,axiom,
    ! [Ma: num] : aa_num_num(bit0,Ma) != one ).

% semiring_norm(85)
tff(fact_5_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_6_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_7_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Ma)),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_8_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_9_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_10_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_11_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xa: nat,Nb: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xa)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_12_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_num(bit0,X2) = aa_num_num(bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_13_semiring__norm_I87_J,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_num(bit0,Ma) = aa_num_num(bit0,Nb) )
    <=> ( Ma = Nb ) ) ).

% semiring_norm(87)
tff(fact_14_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_Extended_enat(numera1916890842035813515d_enat,Ma) = aa_num_Extended_enat(numera1916890842035813515d_enat,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_15_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( numera6690914467698888265omplex(Ma) = numera6690914467698888265omplex(Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_16_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Ma) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_17_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Ma) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_18_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Ma) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_19_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Ma) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_20_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_21_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_22_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(vEBT_vebt_member(Ta),Xa) ) ).

% min_Null_member
tff(fact_23__C2_Ohyps_C_I1_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summary2),m) ).

% "2.hyps"(1)
tff(fact_24_insert_H__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_VEBT_insert(Ta,Xa)),Nb) ) ).

% insert'_pres_valid
tff(fact_25_semiring__norm_I78_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% semiring_norm(78)
tff(fact_26_semiring__norm_I75_J,axiom,
    ! [Ma: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),one) ).

% semiring_norm(75)
tff(fact_27_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_28_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xa)
      <=> aa_set_nat_o(member_nat(Xa),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_29_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_30_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_31_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_32_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_33_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_34_verit__comp__simplify1_I1_J,axiom,
    ! [A: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_35_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,Xa)),Y)
           => ( aa_nat_o(vEBT_vebt_member(Ta),Y)
              | ( Xa = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_36_valid__pres__insert,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_insert(Ta,Xa)),Nb) ) ) ).

% valid_pres_insert
tff(fact_37_enat__ord__number_I2_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Ma)),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_38_aa,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
     => aa_nat_o(vEBT_invar_vebt(X),na) ) ).

% aa
tff(fact_39__C2_Oprems_C_I1_J,axiom,
    aa_nat_o(vEBT_invar_vebt(sa),deg) ).

% "2.prems"(1)
tff(fact_40_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] :
            ( aa_nat_o(aa_nat_fun_nat_o(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_41_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I)
            <=> aa_nat_o(nth_o(Ys),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_42_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),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_43_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] :
            ( aa_nat_o(aa_nat_fun_nat_o(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_44_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
         => ? [X_1: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I),X_1) )
    <=> ? [Xs2: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs2) = K )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I),nth_VEBT_VEBT(Xs2,I)) ) ) ) ).

% Skolem_list_nth
tff(fact_45_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
         => ? [X_1: $o] : aa_o_o(aa_nat_fun_o_o(P,I),(X_1)) )
    <=> ? [Xs2: list_o] :
          ( ( size_size_list_o(Xs2) = K )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
             => aa_o_o(aa_nat_fun_o_o(P,I),aa_nat_o(nth_o(Xs2),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_46_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I),X_1) )
    <=> ? [Xs2: list_nat] :
          ( ( size_size_list_nat(Xs2) = K )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I),aa_nat_nat(nth_nat(Xs2),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_47_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
         => ? [X_1: int] : aa_int_o(aa_nat_fun_int_o(P,I),X_1) )
    <=> ? [Xs2: list_int] :
          ( ( size_size_list_int(Xs2) = K )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
             => aa_int_o(aa_nat_fun_int_o(P,I),aa_nat_int(nth_int(Xs2),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_48_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_49_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_50_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_51_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_52_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_53_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_complex(power_power_complex(numera6690914467698888265omplex(K)),aa_num_nat(numeral_numeral_nat,L)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_54_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_55_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_56_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_57_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_58__C2_OIH_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList2))
     => ( aa_nat_o(vEBT_invar_vebt(X),na)
        & ! [Xa2: vEBT_VEBT] :
            ( aa_nat_o(vEBT_invar_vebt(Xa2),na)
           => ( ( vEBT_VEBT_set_vebt(X) = vEBT_VEBT_set_vebt(Xa2) )
             => ( Xa2 = X ) ) ) ) ) ).

% "2.IH"(1)
tff(fact_59__C2_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "2.hyps"(4)
tff(fact_60_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
    <=> aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq
tff(fact_61_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
     => aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq2
tff(fact_62_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),D)
     => aa_nat_o(vEBT_VEBT_valid(Ta),D) ) ).

% valid_eq1
tff(fact_63_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_64_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),set_complex2(Xs))
         => aa_complex_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_65_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => aa_real_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_66_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Nb: nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),set_set_nat2(Xs))
         => aa_set_nat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,aa_nat_set_nat(nth_set_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_67_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% inthall
tff(fact_68_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),set_o2(Xs))
         => aa_o_o(P,(X3)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_69_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_70_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => aa_int_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_71__C2_OIH_C_I2_J,axiom,
    ! [Sb: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Sb),m)
     => ( ( vEBT_VEBT_set_vebt(summary2) = vEBT_VEBT_set_vebt(Sb) )
       => ( Sb = summary2 ) ) ) ).

% "2.IH"(2)
tff(fact_72_add__numeral__left,axiom,
    ! [V: num,W: num,Z: extended_enat] : plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,V),plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,W),Z)) = plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).

% add_numeral_left
tff(fact_73_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_74_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_75_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_76_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_77_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_78_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Ma),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)) = aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_79_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Ma)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_80_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_81_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_82_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_83_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_84__C2_Ohyps_C_I5_J,axiom,
    ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(summary2),X_12) ).

% "2.hyps"(5)
tff(fact_85__C2_Ohyps_C_I6_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList2))
     => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) ).

% "2.hyps"(6)
tff(fact_86_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% is_num_normalize(1)
tff(fact_87_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_88_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% is_num_normalize(1)
tff(fact_89_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A),collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_90_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( aa_set_real_o(member_real(A),collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_91_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> aa_list_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_92_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(A),collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_93_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_94_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_95_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_96_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_97_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_98_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_99_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_100_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_101_Collect__cong,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X3: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(Q,X3) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_102_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(P,X3)
        <=> aa_list_nat_o(Q,X3) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_103_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
        <=> aa_set_nat_o(Q,X3) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_104_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
        <=> aa_nat_o(Q,X3) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_105_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(Q,X3) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_106_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Nb: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,M),N)
             => aa_Extended_enat_o(P,M) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Nb) ) ).

% enat_less_induct
tff(fact_107_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit0,Nb)) = plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Nb),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)) ).

% numeral_Bit0
tff(fact_108_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_109_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_Bit0
tff(fact_110_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_111_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_112_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_113_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(aa_nat_complex(nth_complex(Xs),Nb)),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_114_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Nb)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_115_nth__mem,axiom,
    ! [Nb: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),Nb)),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_116_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_117_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => aa_set_o_o(member_o(aa_nat_o(nth_o(Xs),Nb)),set_o2(Xs)) ) ).

% nth_mem
tff(fact_118_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_119_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Nb)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_120_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X3: vEBT_VEBT] :
            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X3) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_121_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),set_o2(Xs))
           => aa_o_o(P,(X3)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_122_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
           => aa_nat_o(P,X3) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_123_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),set_int2(Xs))
           => aa_int_o(P,X3) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_124_in__set__conv__nth,axiom,
    ! [Xa: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xa),set_complex2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_125_in__set__conv__nth,axiom,
    ! [Xa: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xa),set_real2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_126_in__set__conv__nth,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3254054031482475050et_nat(Xs))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_127_in__set__conv__nth,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xa,set_VEBT_VEBT2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_128_in__set__conv__nth,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xa)),set_o2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I)
          <=> (Xa) ) ) ) ).

% in_set_conv_nth
tff(fact_129_in__set__conv__nth,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xa),set_nat2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_130_in__set__conv__nth,axiom,
    ! [Xa: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xa),set_int2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_131_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Xa: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I2)) )
     => ( aa_set_complex_o(member_complex(Xa),set_complex2(Xs))
       => aa_complex_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_132_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xa: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) )
     => ( aa_set_real_o(member_real(Xa),set_real2(Xs))
       => aa_real_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_133_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Xa: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P,aa_nat_set_nat(nth_set_nat(Xs),I2)) )
     => ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
       => aa_set_nat_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_134_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xa: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2)) )
     => ( member_VEBT_VEBT(Xa,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_135_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xa: $o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I2)) )
     => ( aa_set_o_o(member_o((Xa)),set_o2(Xs))
       => aa_o_o(P,(Xa)) ) ) ).

% all_nth_imp_all_set
tff(fact_136_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xa: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2)) )
     => ( aa_set_nat_o(member_nat(Xa),set_nat2(Xs))
       => aa_nat_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_137_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xa: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I2)) )
     => ( aa_set_int_o(member_int(Xa),set_int2(Xs))
       => aa_int_o(P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_138_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I)) ) ) ).

% all_set_conv_all_nth
tff(fact_139_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),set_o2(Xs))
         => aa_o_o(P,(X4)) )
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I)) ) ) ).

% all_set_conv_all_nth
tff(fact_140_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_nat_o(P,X4) )
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I)) ) ) ).

% all_set_conv_all_nth
tff(fact_141_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_int_o(P,X4) )
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I)) ) ) ).

% all_set_conv_all_nth
tff(fact_142_pow_Osimps_I1_J,axiom,
    ! [Xa: num] : aa_num_num(aa_num_fun_num_num(pow,Xa),one) = Xa ).

% pow.simps(1)
tff(fact_143_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_144_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_145_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_146_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_147_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs3: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs3) = Nb ).

% Ex_list_of_length
tff(fact_148_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs3: list_o] : size_size_list_o(Xs3) = Nb ).

% Ex_list_of_length
tff(fact_149_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs3: list_nat] : size_size_list_nat(Xs3) = Nb ).

% Ex_list_of_length
tff(fact_150_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs3: list_int] : size_size_list_int(Xs3) = Nb ).

% Ex_list_of_length
tff(fact_151_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs3))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs3) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_152_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs3))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs3) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_153_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs3))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs3) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_154_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs3))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs3) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_155_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_156_nat__add__left__cancel__less,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_157_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_158_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_159_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_160_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_161_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_162_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_163_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_164_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_165_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xa)),Xa) ) ) ).

% valid_insert_both_member_options_add
tff(fact_166_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xa) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_167_ab,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList2))
     => ( vEBT_VEBT_set_vebt(X) = bot_bot_set_nat ) ) ).

% ab
tff(fact_168_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_169__C2_Oprems_C_I2_J,axiom,
    vEBT_VEBT_set_vebt(vEBT_Node(none_P5556105721700978146at_nat,deg,treeList2,summary2)) = vEBT_VEBT_set_vebt(sa) ).

% "2.prems"(2)
tff(fact_170_high__bound__aux,axiom,
    ! [Ma: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_171_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_172_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_173_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_174_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_175_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_176_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_177_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_178_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_179_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_180_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_13) ) ).

% not_min_Null_member
tff(fact_181_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa)
       => aa_nat_o(vEBT_vebt_member(Ta),Xa) ) ) ).

% valid_member_both_member_options
tff(fact_182_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xa) ) ) ).

% both_member_options_equiv_member
tff(fact_183_high__def,axiom,
    ! [Xa: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_184_semiring__norm_I6_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% semiring_norm(6)
tff(fact_185_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_186_sprop,axiom,
    ( ( sa = vEBT_Node(none_P5556105721700978146at_nat,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) )
    & aa_nat_o(vEBT_invar_vebt(summary),m)
    & ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
       => aa_nat_o(vEBT_invar_vebt(X),na) )
    & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(summary),X_12) ) ).

% sprop
tff(fact_187__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_O_As_A_061_ANode_ANone_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_060and_062_A_I_092_060nexists_062i_O_Aboth__member__options_Asummary_H_Ai_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
        ~ ( ( sa = vEBT_Node(none_P5556105721700978146at_nat,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) )
          & aa_nat_o(vEBT_invar_vebt(Summary),m)
          & ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
             => aa_nat_o(vEBT_invar_vebt(X),na) )
          & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_12) ) ).

% \<open>\<And>thesis. (\<And>treeList' summary'. s = Node None 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) \<and> (\<nexists>i. both_member_options summary' i) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_188_power__divide,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_divide
tff(fact_189_power__divide,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_divide
tff(fact_190_power__divide,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),B)),Nb) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_191_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb) = aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one) ).

% add_One_commute
tff(fact_192_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_193_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_194_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_195_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_196_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_197_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_198_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_199_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( ( Ia = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_200_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( ( Ia = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(Ia),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_201_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( ( Ia = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_202_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( ( Ia = J )
        & ( K = L ) )
     => ( aa_int_int(plus_plus_int(Ia),K) = aa_int_int(plus_plus_int(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_203_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_204_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(plus_plus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_205_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_206_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A),B)) ) ) ).

% group_cancel.add1
tff(fact_207_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_208_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_209_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_210_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(plus_plus_int(K),B) )
     => ( aa_int_int(plus_plus_int(A),B2) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A),B)) ) ) ).

% group_cancel.add2
tff(fact_211_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_212_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_213_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_214_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% add.assoc
tff(fact_215_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_216_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_217_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_218_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_219_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_220_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_221_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% add.commute
tff(fact_222_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% add.commute
tff(fact_223_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_224_add_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(B),A) ).

% add.commute
tff(fact_225_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.left_commute
tff(fact_226_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.left_commute
tff(fact_227_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.left_commute
tff(fact_228_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(plus_plus_int(B),aa_int_int(plus_plus_int(A),C)) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% add.left_commute
tff(fact_229_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_230_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_231_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_232_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_233_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_234_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_235_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_236_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_237_linorder__neqE__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_238_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
              & ~ aa_nat_o(P,M) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_239_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
             => aa_nat_o(P,M) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_240_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_241_less__not__refl3,axiom,
    ! [Sb: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Sb),Ta)
     => ( Sb != Ta ) ) ).

% less_not_refl3
tff(fact_242_less__not__refl2,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( Ma != Nb ) ) ).

% less_not_refl2
tff(fact_243_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_244_nat__neq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% nat_neq_iff
tff(fact_245_size__neq__size__imp__neq,axiom,
    ! [Xa: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xa) != size_s6755466524823107622T_VEBT(Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_246_size__neq__size__imp__neq,axiom,
    ! [Xa: list_o,Y: list_o] :
      ( ( size_size_list_o(Xa) != size_size_list_o(Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_247_size__neq__size__imp__neq,axiom,
    ! [Xa: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(Xa) != size_size_list_nat(Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_248_size__neq__size__imp__neq,axiom,
    ! [Xa: list_int,Y: list_int] :
      ( ( size_size_list_int(Xa) != size_size_list_int(Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_249_size__neq__size__imp__neq,axiom,
    ! [Xa: num,Y: num] :
      ( ( aa_num_nat(size_size_num,Xa) != aa_num_nat(size_size_num,Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_250_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_251_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_252_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_253_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_254_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_255_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_256_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_257_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_258_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_strict_right_mono
tff(fact_259_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_strict_right_mono
tff(fact_260_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_261_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)) ) ).

% add_strict_right_mono
tff(fact_262_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_strict_left_mono
tff(fact_263_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_strict_left_mono
tff(fact_264_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_265_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)) ) ).

% add_strict_left_mono
tff(fact_266_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_267_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_268_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_269_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) ) ) ).

% add_strict_mono
tff(fact_270_add__strict__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),D)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,plus_p3455044024723400733d_enat(A,C)),plus_p3455044024723400733d_enat(B,D)) ) ) ).

% add_strict_mono
tff(fact_271_add__mono__thms__linordered__field_I1_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ia),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_272_add__mono__thms__linordered__field_I1_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ia),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Ia),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_273_add__mono__thms__linordered__field_I1_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_274_add__mono__thms__linordered__field_I1_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Ia),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_275_add__mono__thms__linordered__field_I2_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( ( Ia = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_276_add__mono__thms__linordered__field_I2_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( ( Ia = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Ia),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_277_add__mono__thms__linordered__field_I2_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( ( Ia = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_278_add__mono__thms__linordered__field_I2_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( ( Ia = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Ia),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_279_add__mono__thms__linordered__field_I5_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ia),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_280_add__mono__thms__linordered__field_I5_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ia),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Ia),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_281_add__mono__thms__linordered__field_I5_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_282_add__mono__thms__linordered__field_I5_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Ia),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_283_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% less_add_eq_less
tff(fact_284_trans__less__add2,axiom,
    ! [Ia: nat,J: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),J)) ) ).

% trans_less_add2
tff(fact_285_trans__less__add1,axiom,
    ! [Ia: nat,J: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ma)) ) ).

% trans_less_add1
tff(fact_286_add__less__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_287_not__add__less2,axiom,
    ! [J: nat,Ia: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ia)),Ia) ).

% not_add_less2
tff(fact_288_not__add__less1,axiom,
    ! [Ia: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J)),Ia) ).

% not_add_less1
tff(fact_289_add__less__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_290_add__lessD1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),K) ) ).

% add_lessD1
tff(fact_291_add__self__div__2,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Ma)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Ma ).

% add_self_div_2
tff(fact_292_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Ma: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Ma)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
         => ( ( Ma = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma) )
             => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_293_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xa) ) ) ) ).

% both_member_options_ding
tff(fact_294_div__exp__eq,axiom,
    ! [A: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ).

% div_exp_eq
tff(fact_295_div__exp__eq,axiom,
    ! [A: int,Ma: nat,Nb: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ).

% div_exp_eq
tff(fact_296_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_297_field__less__half__sum,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_298_field__less__half__sum,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xa),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_299_high__inv,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xa)),Nb) = Y ) ) ).

% high_inv
tff(fact_300_field__sum__of__halves,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xa ).

% field_sum_of_halves
tff(fact_301_field__sum__of__halves,axiom,
    ! [Xa: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xa),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(Xa),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xa ).

% field_sum_of_halves
tff(fact_302_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit0_div_2
tff(fact_303_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_304_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_305_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),Xa) ).

% vebt_member.simps(2)
tff(fact_306_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_307_bit__split__inv,axiom,
    ! [Xa: 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,Xa),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),D)),D) = Xa ).

% bit_split_inv
tff(fact_308_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: extended_enat] : times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,V),times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,W),Z)) = times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_309_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_310_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_311_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_312_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_313_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_314_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Ma),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)) = aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% numeral_times_numeral
tff(fact_315_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Ma)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% numeral_times_numeral
tff(fact_316_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% numeral_times_numeral
tff(fact_317_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% numeral_times_numeral
tff(fact_318_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% numeral_times_numeral
tff(fact_319_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% numeral_times_numeral
tff(fact_320_low__inv,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xa)),Nb) = Xa ) ) ).

% low_inv
tff(fact_321_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_322_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] : times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(A,B),aa_num_Extended_enat(numera1916890842035813515d_enat,V)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(A,aa_num_Extended_enat(numera1916890842035813515d_enat,V)),times_7803423173614009249d_enat(B,aa_num_Extended_enat(numera1916890842035813515d_enat,V))) ).

% distrib_right_numeral
tff(fact_323_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_324_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_325_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_326_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_327_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_328_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] : times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,V),plus_p3455044024723400733d_enat(B,C)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,V),B),times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,V),C)) ).

% distrib_left_numeral
tff(fact_329_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_330_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_331_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_332_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_333_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_334_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_335_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_336_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_337_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_338_power__add__numeral,axiom,
    ! [A: complex,Ma: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% power_add_numeral
tff(fact_339_power__add__numeral,axiom,
    ! [A: real,Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% power_add_numeral
tff(fact_340_power__add__numeral,axiom,
    ! [A: rat,Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% power_add_numeral
tff(fact_341_power__add__numeral,axiom,
    ! [A: nat,Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% power_add_numeral
tff(fact_342_power__add__numeral,axiom,
    ! [A: int,Ma: num,Nb: num] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% power_add_numeral
tff(fact_343_power__add__numeral2,axiom,
    ! [A: complex,Ma: num,Nb: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_344_power__add__numeral2,axiom,
    ! [A: real,Ma: num,Nb: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_345_power__add__numeral2,axiom,
    ! [A: rat,Ma: num,Nb: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_346_power__add__numeral2,axiom,
    ! [A: nat,Ma: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_347_power__add__numeral2,axiom,
    ! [A: int,Ma: num,Nb: num,B: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_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,Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_348_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_349_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_350_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_351_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_352_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_353_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_354_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_355_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% mult.assoc
tff(fact_356_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_357_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_358_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_359_mult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(A),B) = aa_int_int(times_times_int(B),A) ).

% mult.commute
tff(fact_360_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.left_commute
tff(fact_361_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.left_commute
tff(fact_362_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.left_commute
tff(fact_363_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(times_times_int(B),aa_int_int(times_times_int(A),C)) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% mult.left_commute
tff(fact_364_div__mult2__eq,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Q2) ).

% div_mult2_eq
tff(fact_365_power__commutes,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_commutes
tff(fact_366_power__commutes,axiom,
    ! [A: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_commutes
tff(fact_367_power__commutes,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_commutes
tff(fact_368_power__commutes,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_commutes
tff(fact_369_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),Nb)),A) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb)) ).

% power_commutes
tff(fact_370_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_mult_distrib
tff(fact_371_power__mult__distrib,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_mult_distrib
tff(fact_372_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_mult_distrib
tff(fact_373_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_374_power__mult__distrib,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(times_times_int(A),B)),Nb) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ).

% power_mult_distrib
tff(fact_375_power__commuting__commutes,axiom,
    ! [Xa: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xa),Y) = aa_complex_complex(times_times_complex(Y),Xa) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xa),Nb)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(Xa),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_376_power__commuting__commutes,axiom,
    ! [Xa: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xa) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_377_power__commuting__commutes,axiom,
    ! [Xa: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xa),Y) = aa_rat_rat(times_times_rat(Y),Xa) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xa),Nb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(Xa),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_378_power__commuting__commutes,axiom,
    ! [Xa: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xa) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Nb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_379_power__commuting__commutes,axiom,
    ! [Xa: int,Y: int,Nb: nat] :
      ( ( aa_int_int(times_times_int(Xa),Y) = aa_int_int(times_times_int(Y),Xa) )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(Xa),Nb)),Y) = aa_int_int(times_times_int(Y),aa_nat_int(power_power_int(Xa),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_380_power__mult,axiom,
    ! [A: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),Nb) ).

% power_mult
tff(fact_381_power__mult,axiom,
    ! [A: real,Ma: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),Nb) ).

% power_mult
tff(fact_382_power__mult,axiom,
    ! [A: int,Ma: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Ma)),Nb) ).

% power_mult
tff(fact_383_power__mult,axiom,
    ! [A: complex,Ma: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Ma)),Nb) ).

% power_mult
tff(fact_384_left__add__mult__distrib,axiom,
    ! [Ia: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_385_add__mult__distrib2,axiom,
    ! [K: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% add_mult_distrib2
tff(fact_386_add__mult__distrib,axiom,
    ! [Ma: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% add_mult_distrib
tff(fact_387_less__mult__imp__div__less,axiom,
    ! [Ma: nat,Ia: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Ia) ) ).

% less_mult_imp_div_less
tff(fact_388_mult__numeral__1,axiom,
    ! [A: extended_enat] : times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,one),A) = A ).

% mult_numeral_1
tff(fact_389_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_390_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_391_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_392_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_393_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_394_mult__numeral__1__right,axiom,
    ! [A: extended_enat] : times_7803423173614009249d_enat(A,aa_num_Extended_enat(numera1916890842035813515d_enat,one)) = A ).

% mult_numeral_1_right
tff(fact_395_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_396_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_397_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_398_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_399_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_400_power__add,axiom,
    ! [A: complex,Ma: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Ma)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_add
tff(fact_401_power__add,axiom,
    ! [A: real,Ma: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_402_power__add,axiom,
    ! [A: rat,Ma: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Ma)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_403_power__add,axiom,
    ! [A: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_404_power__add,axiom,
    ! [A: int,Ma: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),Ma)),aa_nat_int(power_power_int(A),Nb)) ).

% power_add
tff(fact_405_mult__2,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit0,one)),Z) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2
tff(fact_406_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_407_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_408_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_409_mult__2,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2
tff(fact_410_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_411_mult__2__right,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(Z,aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2_right
tff(fact_412_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_413_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_414_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_415_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2_right
tff(fact_416_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_417_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(aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit0,one)),A),B) ).

% left_add_twice
tff(fact_418_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_419_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_420_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_421_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_422_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_423_power4__eq__xxxx,axiom,
    ! [Xa: complex] : aa_nat_complex(power_power_complex(Xa),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(Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_424_power4__eq__xxxx,axiom,
    ! [Xa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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,Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_425_power4__eq__xxxx,axiom,
    ! [Xa: rat] : aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_426_power4__eq__xxxx,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),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,Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_427_power4__eq__xxxx,axiom,
    ! [Xa: int] : aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(times_times_int(Xa),Xa)),Xa)),Xa) ).

% power4_eq_xxxx
tff(fact_428_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_429_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_430_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A),A) ).

% power2_eq_square
tff(fact_431_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_432_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(times_times_int(A),A) ).

% power2_eq_square
tff(fact_433_power__even__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_434_power__even__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_435_power__even__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_436_power__even__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_437_power2__sum,axiom,
    ! [Xa: extended_enat,Y: extended_enat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(plus_p3455044024723400733d_enat(Xa,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(power_8040749407984259932d_enat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_7803423173614009249d_enat(times_7803423173614009249d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit0,one)),Xa),Y)) ).

% power2_sum
tff(fact_438_power2__sum,axiom,
    ! [Xa: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xa),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(Xa),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))),Xa)),Y)) ).

% power2_sum
tff(fact_439_power2__sum,axiom,
    ! [Xa: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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))),Xa)),Y)) ).

% power2_sum
tff(fact_440_power2__sum,axiom,
    ! [Xa: 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,Xa),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,Xa),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))),Xa)),Y)) ).

% power2_sum
tff(fact_441_power2__sum,axiom,
    ! [Xa: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(plus_plus_int(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xa),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(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_sum
tff(fact_442_power2__sum,axiom,
    ! [Xa: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_sum
tff(fact_443_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xa)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Nb)) ) ).

% in_children_def
tff(fact_444_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_445_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).

% times_divide_eq_right
tff(fact_446_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_447_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),C) ).

% times_divide_eq_right
tff(fact_448_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).

% divide_divide_eq_right
tff(fact_449_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_450_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),B) ).

% divide_divide_eq_right
tff(fact_451_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_452_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_453_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_454_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).

% times_divide_eq_left
tff(fact_455_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_456_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),A)),C) ).

% times_divide_eq_left
tff(fact_457_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xa: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),Xa) ).

% buildup_nothing_in_leaf
tff(fact_458_low__def,axiom,
    ! [Xa: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_459_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Ma: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Ma)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
         => ( ( Ma = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma) )
             => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_460_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_461_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu,none_P5556105721700978146at_nat,Uv) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_462_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_num_fun_num_num,Uv: option_num] : vEBT_V819420779217536731ft_num(Uu,none_num,Uv) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_463_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu),none_nat),Uv) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_464_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_465_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_466_empty__Collect__eq,axiom,
    ! [P: fun_real_o] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X4: real] : ~ aa_real_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_467_empty__Collect__eq,axiom,
    ! [P: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(P) )
    <=> ! [X4: $o] : ~ aa_o_o(P,(X4)) ) ).

% empty_Collect_eq
tff(fact_468_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_469_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_470_even__odd__cases,axiom,
    ! [Xa: nat] :
      ( ! [N: nat] : Xa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : Xa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_471_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_472_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_473_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_474_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList2,S) ) ).

% deg_SUcn_Node
tff(fact_475_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_476_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_477_empty__iff,axiom,
    ! [C: complex] : ~ aa_set_complex_o(member_complex(C),bot_bot_set_complex) ).

% empty_iff
tff(fact_478_empty__iff,axiom,
    ! [C: set_nat] : ~ aa_set_set_nat_o(member_set_nat(C),bot_bot_set_set_nat) ).

% empty_iff
tff(fact_479_empty__iff,axiom,
    ! [C: real] : ~ aa_set_real_o(member_real(C),bot_bot_set_real) ).

% empty_iff
tff(fact_480_empty__iff,axiom,
    ! [C: $o] : ~ aa_set_o_o(member_o((C)),bot_bot_set_o) ).

% empty_iff
tff(fact_481_empty__iff,axiom,
    ! [C: nat] : ~ aa_set_nat_o(member_nat(C),bot_bot_set_nat) ).

% empty_iff
tff(fact_482_empty__iff,axiom,
    ! [C: int] : ~ aa_set_int_o(member_int(C),bot_bot_set_int) ).

% empty_iff
tff(fact_483_all__not__in__conv,axiom,
    ! [A2: set_complex] :
      ( ! [X4: complex] : ~ aa_set_complex_o(member_complex(X4),A2)
    <=> ( A2 = bot_bot_set_complex ) ) ).

% all_not_in_conv
tff(fact_484_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X4),A2)
    <=> ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_485_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] : ~ aa_set_real_o(member_real(X4),A2)
    <=> ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_486_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ! [X4: $o] : ~ aa_set_o_o(member_o((X4)),A2)
    <=> ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_487_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(member_nat(X4),A2)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_488_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int(X4),A2)
    <=> ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_489_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_490_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_491_Collect__empty__eq,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X4: real] : ~ aa_real_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_492_Collect__empty__eq,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ! [X4: $o] : ~ aa_o_o(P,(X4)) ) ).

% Collect_empty_eq
tff(fact_493_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_494_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_495_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_496_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mod_trivial
tff(fact_497_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_mod_trivial
tff(fact_498_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_499_not__gr__zero,axiom,
    ! [Nb: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% not_gr_zero
tff(fact_500_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_501_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_502_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_503_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_504_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_505_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_506_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_507_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(plus_plus_int(A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_508_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_509_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_510_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_511_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_512_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_513_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_514_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_515_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_516_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_517_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_518_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_519_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_520_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_521_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_522_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_523_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_524_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_525_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_526_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_527_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_528_add__eq__0__iff__both__eq__0,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
    <=> ( ( Xa = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_529_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) )
    <=> ( ( Xa = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_530_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_531_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_532_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_533_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_534_add__0,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% add_0
tff(fact_535_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_536_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_537_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_538_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_539_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_540_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_541_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_542_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_543_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_544_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_545_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_546_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_547_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_548_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_549_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_550_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_551_Suc__less__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_less_eq
tff(fact_552_Suc__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_553_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_554_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_555_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_556_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_557_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_558_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_559_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_560_add__Suc__right,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% add_Suc_right
tff(fact_561_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_562_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self1
tff(fact_563_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_564_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_565_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self2
tff(fact_566_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_567_add__is__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) = zero_zero_nat )
    <=> ( ( Ma = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_568_Nat_Oadd__0__right,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),zero_zero_nat) = Ma ).

% Nat.add_0_right
tff(fact_569_mult__cancel2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K) )
    <=> ( ( Ma = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_570_mult__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( Ma = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_571_mult__0__right,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_572_mult__is__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) = zero_zero_nat )
    <=> ( ( Ma = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_573_mod__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = Ma ) ) ).

% mod_less
tff(fact_574_semiring__norm_I13_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% semiring_norm(13)
tff(fact_575_semiring__norm_I11_J,axiom,
    ! [Ma: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Ma),one) = Ma ).

% semiring_norm(11)
tff(fact_576_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_577_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_578_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_579_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_580_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_581_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_582_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_583_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_584_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_585_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_586_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(plus_plus_int(B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_587_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_588_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_589_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_590_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(plus_plus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_591_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_592_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_593_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_594_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_595_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_596_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_597_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_598_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_599_sum__squares__eq__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_600_sum__squares__eq__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xa),Xa)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_601_sum__squares__eq__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xa),Xa)),aa_int_int(times_times_int(Y),Y)) = zero_zero_int )
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_602_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_603_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_604_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_605_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_606_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_607_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_608_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_609_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_610_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_611_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_612_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_613_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_614_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_615_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_616_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_617_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_618_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(divide_divide_int(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(divide_divide_int(A),B)) ).

% div_mult_mult1_if
tff(fact_619_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_620_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_621_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_622_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_623_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_624_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_625_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_626_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_627_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_628_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_629_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_630_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_631_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_632_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_633_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_634_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_635_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_636_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_637_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_638_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_639_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_640_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_641_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_642_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_643_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_644_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_645_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_646_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_647_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_648_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_649_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_650_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),B))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_651_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_652_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_653_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(B),C))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_654_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_655_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_656_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),B)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_657_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_658_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_659_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),C)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_660_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_661_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_662_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_663_add__gr__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_664_one__eq__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) )
    <=> ( ( Ma = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_665_mult__eq__1__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Ma = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_666_div__by__Suc__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(suc,zero_zero_nat)) = Ma ).

% div_by_Suc_0
tff(fact_667_mult__less__cancel2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% mult_less_cancel2
tff(fact_668_nat__0__less__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_669_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_670_div__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_671_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_672_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Ma) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Ma = zero_zero_nat )
        | ( Xa = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_673_mult__Suc__right,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ).

% mult_Suc_right
tff(fact_674_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_675_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_676_mod__by__Suc__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_677_num__double,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_678_power__mult__numeral,axiom,
    ! [A: nat,Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% power_mult_numeral
tff(fact_679_power__mult__numeral,axiom,
    ! [A: real,Ma: num,Nb: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% power_mult_numeral
tff(fact_680_power__mult__numeral,axiom,
    ! [A: int,Ma: num,Nb: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% power_mult_numeral
tff(fact_681_power__mult__numeral,axiom,
    ! [A: complex,Ma: num,Nb: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% power_mult_numeral
tff(fact_682_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_683_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_684_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_685_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_686_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_687_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_688_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_689_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),C)),A)),B) = aa_int_int(plus_plus_int(C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_690_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_691_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),B)),A)),B) = aa_int_int(plus_plus_int(C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_692_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_693_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(B),C))),B) = aa_int_int(plus_plus_int(C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_694_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_695_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),B))),B) = aa_int_int(plus_plus_int(C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_696_power__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_697_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_698_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_699_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(power_power_int(A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_700_power__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Nb) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_701_div__mult__self__is__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),Nb) = Ma ) ) ).

% div_mult_self_is_m
tff(fact_702_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Ma)),Nb) = Ma ) ) ).

% div_mult_self1_is_m
tff(fact_703_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% Suc_numeral
tff(fact_704_Suc__mod__mult__self1,axiom,
    ! [Ma: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% Suc_mod_mult_self1
tff(fact_705_Suc__mod__mult__self2,axiom,
    ! [Ma: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% Suc_mod_mult_self2
tff(fact_706_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),Ma))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% Suc_mod_mult_self3
tff(fact_707_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)),Ma))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% Suc_mod_mult_self4
tff(fact_708_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_709_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_710_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_711_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_712_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_713_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_714_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_715_div2__Suc__Suc,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Ma))),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,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_716_mod2__Suc__Suc,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_717_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_718_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_719_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_720_sum__power2__eq__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xa),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 )
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_721_sum__power2__eq__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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 )
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_722_sum__power2__eq__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xa),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 )
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_723_add__self__mod__2,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Ma)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_724_mod__Suc__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Ma))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_725_mod__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% mod_Suc_eq
tff(fact_726_mod__Suc,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))) ).

% mod_Suc
tff(fact_727_vebt__buildup_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero_nat )
     => ( ( Xa != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : Xa != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_728_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% not0_implies_Suc
tff(fact_729_zero__reorient,axiom,
    ! [Xa: complex] :
      ( ( zero_zero_complex = Xa )
    <=> ( Xa = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_730_zero__reorient,axiom,
    ! [Xa: real] :
      ( ( zero_zero_real = Xa )
    <=> ( Xa = zero_zero_real ) ) ).

% zero_reorient
tff(fact_731_zero__reorient,axiom,
    ! [Xa: rat] :
      ( ( zero_zero_rat = Xa )
    <=> ( Xa = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_732_zero__reorient,axiom,
    ! [Xa: nat] :
      ( ( zero_zero_nat = Xa )
    <=> ( Xa = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_733_zero__reorient,axiom,
    ! [Xa: int] :
      ( ( zero_zero_int = Xa )
    <=> ( Xa = zero_zero_int ) ) ).

% zero_reorient
tff(fact_734_Zero__not__Suc,axiom,
    ! [Ma: nat] : zero_zero_nat != aa_nat_nat(suc,Ma) ).

% Zero_not_Suc
tff(fact_735_Zero__neq__Suc,axiom,
    ! [Ma: nat] : zero_zero_nat != aa_nat_nat(suc,Ma) ).

% Zero_neq_Suc
tff(fact_736_Suc__neq__Zero,axiom,
    ! [Ma: nat] : aa_nat_nat(suc,Ma) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_737_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_738_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_739_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Ma: nat,Nb: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X3),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Ma),Nb) ) ) ) ).

% diff_induct
tff(fact_740_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_741_Suc__inject,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xa) = aa_nat_nat(suc,Y) )
     => ( Xa = Y ) ) ).

% Suc_inject
tff(fact_742_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_743_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_744_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_745_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_746_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_747_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Ma: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),P2)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P2)) ) )
           => aa_nat_o(P,Ma) ) ) ) ) ).

% mod_induct
tff(fact_748_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_749_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_750_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_751_mod__less__divisor,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_752_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I)) ) ) ) ).

% Ex_less_Suc2
tff(fact_753_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M3: nat] : Nb = aa_nat_nat(suc,M3) ) ).

% gr0_conv_Suc
tff(fact_754_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I)) ) ) ) ).

% All_less_Suc2
tff(fact_755_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% gr0_implies_Suc
tff(fact_756_less__Suc__eq__0__disj,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb))
    <=> ( ( Ma = zero_zero_nat )
        | ? [J2: nat] :
            ( ( Ma = aa_nat_nat(suc,J2) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_757_one__is__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) )
    <=> ( ( ( Ma = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Ma = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_758_add__is__1,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Ma = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Ma = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_759_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_760_div__less__mono,axiom,
    ! [A2: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Nb) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_761_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_762_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_763_n__less__n__mult__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Ma)) ) ) ).

% n_less_n_mult_m
tff(fact_764_n__less__m__mult__n,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_765_one__less__mult,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ) ) ).

% one_less_mult
tff(fact_766_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K)) ) ).

% power_gt_expt
tff(fact_767_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_768_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_769_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_770_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(power_power_int(zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_771_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_772_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_773_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_eq
tff(fact_774_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_eq
tff(fact_775_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_776_mod__mult__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A3),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_777_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A3),B3),C) ) ) ) ).

% mod_mult_cong
tff(fact_778_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_779_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),B)),C) ).

% mod_mult_mult2
tff(fact_780_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,B)),C) ).

% mod_mult_mult2
tff(fact_781_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_782_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(times_times_int(C),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ).

% mult_mod_right
tff(fact_783_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).

% mult_mod_right
tff(fact_784_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_785_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_left_eq
tff(fact_786_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_left_eq
tff(fact_787_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_788_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_right_eq
tff(fact_789_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_right_eq
tff(fact_790_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_791_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_eq
tff(fact_792_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_793_mod__add__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_794_mod__add__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A3),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_795_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A3,B3),C) ) ) ) ).

% mod_add_cong
tff(fact_796_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_797_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_left_eq
tff(fact_798_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_799_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_800_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_right_eq
tff(fact_801_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_802_power__mod,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).

% power_mod
tff(fact_803_power__mod,axiom,
    ! [A: int,B: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_int_int(modulo_modulo_int(A),B)),Nb)),B) = aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),Nb)),B) ).

% power_mod
tff(fact_804_power__mod,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),Nb),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),B) ).

% power_mod
tff(fact_805_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma))
      <=> ( Nb = Ma ) ) ) ).

% not_less_less_Suc_eq
tff(fact_806_strict__inc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I2))
               => aa_nat_o(P,I2) ) )
         => aa_nat_o(P,Ia) ) ) ) ).

% strict_inc_induct
tff(fact_807_less__Suc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J3: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J3),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Ia),J) ) ) ) ).

% less_Suc_induct
tff(fact_808_less__trans__Suc,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ia)),K) ) ) ).

% less_trans_Suc
tff(fact_809_Suc__less__SucD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_less_SucD
tff(fact_810_less__antisym,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma))
       => ( Ma = Nb ) ) ) ).

% less_antisym
tff(fact_811_Suc__less__eq2,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Ma)
    <=> ? [M4: nat] :
          ( ( Ma = aa_nat_nat(suc,M4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M4) ) ) ).

% Suc_less_eq2
tff(fact_812_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
           => aa_nat_o(P,I) ) ) ) ).

% All_less_Suc
tff(fact_813_not__less__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma)) ) ).

% not_less_eq
tff(fact_814_less__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Ma = Nb ) ) ) ).

% less_Suc_eq
tff(fact_815_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
            & aa_nat_o(P,I) ) ) ) ).

% Ex_less_Suc
tff(fact_816_less__SucI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_817_less__SucE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => ( Ma = Nb ) ) ) ).

% less_SucE
tff(fact_818_Suc__lessI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( ( aa_nat_nat(suc,Ma) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),Nb) ) ) ).

% Suc_lessI
tff(fact_819_Suc__lessE,axiom,
    ! [Ia: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ia)),K)
     => ~ ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J3)
           => ( K != aa_nat_nat(suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_820_Suc__lessD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_lessD
tff(fact_821_Nat_OlessE,axiom,
    ! [Ia: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),K)
     => ( ( K != aa_nat_nat(suc,Ia) )
       => ~ ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J3)
             => ( K != aa_nat_nat(suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_822_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_823_add__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% add_Suc
tff(fact_824_add__Suc__shift,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_825_Suc__mult__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb) )
    <=> ( Ma = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_826_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_827_zero__less__iff__neq__zero,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb != zero_z5237406670263579293d_enat ) ) ).

% zero_less_iff_neq_zero
tff(fact_828_gr__implies__not__zero,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_829_gr__implies__not__zero,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Ma),Nb)
     => ( Nb != zero_z5237406670263579293d_enat ) ) ).

% gr_implies_not_zero
tff(fact_830_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_831_not__less__zero,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Nb),zero_z5237406670263579293d_enat) ).

% not_less_zero
tff(fact_832_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_833_gr__zeroI,axiom,
    ! [Nb: extended_enat] :
      ( ( Nb != zero_z5237406670263579293d_enat )
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Nb) ) ).

% gr_zeroI
tff(fact_834_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_835_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_836_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_837_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_838_less__numeral__extra_I3_J,axiom,
    ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat) ).

% less_numeral_extra(3)
tff(fact_839_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
       => ? [E: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D2) ) ) ) ).

% field_lbound_gt_zero
tff(fact_840_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D2)
       => ? [E: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D2) ) ) ) ).

% field_lbound_gt_zero
tff(fact_841_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_z5237406670263579293d_enat != aa_num_Extended_enat(numera1916890842035813515d_enat,Nb) ).

% zero_neq_numeral
tff(fact_842_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_843_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_844_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_845_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_846_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_847_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_848_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_849_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_850_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_851_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_852_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_853_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_854_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_855_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_856_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_857_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_858_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_859_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_860_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_861_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_862_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_863_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_864_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_865_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_866_power__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_867_power__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_868_power__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_869_power__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_870_power__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_871_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_872_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
                  & ~ aa_nat_o(P,M) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_873_gr__implies__not0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_874_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_875_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_876_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_877_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_878_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_879_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_880_add__eq__self__zero,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) = Ma )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_881_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( Ma = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_882_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_883_divide__neg__neg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% divide_neg_neg
tff(fact_884_divide__neg__neg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% divide_neg_neg
tff(fact_885_divide__neg__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_886_divide__neg__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_887_divide__pos__neg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_888_divide__pos__neg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_889_divide__pos__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% divide_pos_pos
tff(fact_890_divide__pos__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% divide_pos_pos
tff(fact_891_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_892_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_893_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_894_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_895_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_896_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_897_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_898_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_899_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_900_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_901_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_902_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_903_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_904_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_905_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_906_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_907_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_908_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_909_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_910_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_911_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_912_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_913_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_914_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_915_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_916_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_917_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_918_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_919_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,Xa: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xa),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
        <=> ( aa_complex_complex(times_times_complex(Xa),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_920_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_921_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(Xa),Y) = aa_rat_rat(divide_divide_rat(W),Z) )
        <=> ( aa_rat_rat(times_times_rat(Xa),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_922_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_923_split__mod,axiom,
    ! [P: fun_nat_o,Ma: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Ma) )
        & ( ( Nb != zero_zero_nat )
         => ! [I: nat,J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => ( ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I)),J2) )
               => aa_nat_o(P,J2) ) ) ) ) ) ).

% split_mod
tff(fact_924_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D3: int] : B != aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),D3)) ) ).

% mod_eqE
tff(fact_925_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D3: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D3)) ) ).

% mod_eqE
tff(fact_926_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_927_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_928_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_929_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_930_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_931_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_932_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_933_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_934_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_935_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_936_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_937_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_938_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_na8797796813480665199d_enat,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,N)),aa_nat_Extended_enat(F,aa_nat_nat(suc,N)))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,Nb)),aa_nat_Extended_enat(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_939_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_940_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_941_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_942_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_943_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_944_lift__Suc__mono__less,axiom,
    ! [F: fun_na8797796813480665199d_enat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,N)),aa_nat_Extended_enat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,Nb)),aa_nat_Extended_enat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_945_power__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_Suc
tff(fact_946_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_Suc
tff(fact_947_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_948_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_949_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb)) ).

% power_Suc
tff(fact_950_power__Suc2,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) ).

% power_Suc2
tff(fact_951_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) ).

% power_Suc2
tff(fact_952_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_953_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_954_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),Nb)),A) ).

% power_Suc2
tff(fact_955_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_956_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_957_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_958_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_959_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_960_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_961_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_962_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_963_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_964_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_965_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_966_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_967_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xa: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_968_mult__imp__div__pos__less,axiom,
    ! [Y: rat,Xa: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_969_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_970_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_971_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_972_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_973_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_974_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_975_divide__add__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xa),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_976_divide__add__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_977_divide__add__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xa),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xa),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_978_add__divide__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xa),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xa),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_979_add__divide__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_980_add__divide__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Xa),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xa),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_981_add__num__frac,axiom,
    ! [Y: complex,Z: complex,Xa: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xa),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xa),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_982_add__num__frac,axiom,
    ! [Y: real,Z: real,Xa: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_983_add__num__frac,axiom,
    ! [Y: rat,Z: rat,Xa: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(Xa),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xa),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_984_add__frac__num,axiom,
    ! [Y: complex,Xa: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xa),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_985_add__frac__num,axiom,
    ! [Y: real,Xa: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_986_add__frac__num,axiom,
    ! [Y: rat,Xa: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xa),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xa),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_987_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xa: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xa),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_988_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_989_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xa),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xa),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_990_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_991_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_992_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_993_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_994_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_995_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_996_less__imp__Suc__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K2)) ) ).

% less_imp_Suc_add
tff(fact_997_less__iff__Suc__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K3)) ) ).

% less_iff_Suc_add
tff(fact_998_less__add__Suc2,axiom,
    ! [Ia: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Ia))) ).

% less_add_Suc2
tff(fact_999_less__add__Suc1,axiom,
    ! [Ia: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Ma))) ).

% less_add_Suc1
tff(fact_1000_less__natE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ~ ! [Q3: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Q3)) ) ).

% less_natE
tff(fact_1001_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_1002_mult__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ).

% mult_Suc
tff(fact_1003_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)),zero_z5237406670263579293d_enat) ).

% not_numeral_less_zero
tff(fact_1004_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_1005_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_1006_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_1007_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_1008_zero__less__numeral,axiom,
    ! [Nb: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)) ).

% zero_less_numeral
tff(fact_1009_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_less_numeral
tff(fact_1010_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_1011_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_1012_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_less_numeral
tff(fact_1013_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_1014_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% pos_add_strict
tff(fact_1015_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_1016_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% pos_add_strict
tff(fact_1017_pos__add__strict,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),plus_p3455044024723400733d_enat(A,C)) ) ) ).

% pos_add_strict
tff(fact_1018_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C2: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1019_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ~ ! [C2: extended_enat] :
            ( ( B = plus_p3455044024723400733d_enat(A,C2) )
           => ( C2 = zero_z5237406670263579293d_enat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1020_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_1021_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_pos
tff(fact_1022_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_1023_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B)) ) ) ).

% add_pos_pos
tff(fact_1024_add__pos__pos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),plus_p3455044024723400733d_enat(A,B)) ) ) ).

% add_pos_pos
tff(fact_1025_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_1026_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_1027_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_1028_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_1029_add__neg__neg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),zero_z5237406670263579293d_enat)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,plus_p3455044024723400733d_enat(A,B)),zero_z5237406670263579293d_enat) ) ) ).

% add_neg_neg
tff(fact_1030_zero__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_less_power
tff(fact_1031_zero__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_less_power
tff(fact_1032_zero__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_less_power
tff(fact_1033_zero__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_less_power
tff(fact_1034_less__imp__add__positive,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1035_mult__less__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_1036_mult__less__mono2,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ia)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_1037_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1038_nat__mult__less__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_1039_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1040_nat__power__less__imp__less,axiom,
    ! [Ia: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ia)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_1041_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_1042_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(plus_plus_int(A),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_1043_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_1044_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_1045_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1046_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_1047_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_1048_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_1049_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_1050_odd__power__less__zero,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_1051_odd__power__less__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_1052_odd__power__less__zero,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_1053_div__mod__decomp,axiom,
    ! [A2: nat,Nb: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb)) ).

% div_mod_decomp
tff(fact_1054_mod__mult2__eq,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) ).

% mod_mult2_eq
tff(fact_1055_linordered__field__no__ub,axiom,
    ! [X: real] :
    ? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),X_13) ).

% linordered_field_no_ub
tff(fact_1056_linordered__field__no__ub,axiom,
    ! [X: rat] :
    ? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X_13) ).

% linordered_field_no_ub
tff(fact_1057_linordered__field__no__lb,axiom,
    ! [X: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X) ).

% linordered_field_no_lb
tff(fact_1058_linordered__field__no__lb,axiom,
    ! [X: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X) ).

% linordered_field_no_lb
tff(fact_1059_sum__squares__gt__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
    <=> ( ( Xa != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1060_sum__squares__gt__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xa),Xa)),aa_rat_rat(times_times_rat(Y),Y)))
    <=> ( ( Xa != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1061_sum__squares__gt__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xa),Xa)),aa_int_int(times_times_int(Y),Y)))
    <=> ( ( Xa != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1062_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1063_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_num_real(numeral_numeral_real,W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C),aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1064_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C),aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1065_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1066_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C) = B,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1067_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) = B,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1068_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ aa_set_real_o(ord_less_set_real(A2),bot_bot_set_real) ).

% not_psubset_empty
tff(fact_1069_not__psubset__empty,axiom,
    ! [A2: set_o] : ~ aa_set_o_o(ord_less_set_o(A2),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_1070_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A2),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_1071_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(ord_less_set_int(A2),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_1072_emptyE,axiom,
    ! [A: complex] : ~ aa_set_complex_o(member_complex(A),bot_bot_set_complex) ).

% emptyE
tff(fact_1073_emptyE,axiom,
    ! [A: set_nat] : ~ aa_set_set_nat_o(member_set_nat(A),bot_bot_set_set_nat) ).

% emptyE
tff(fact_1074_emptyE,axiom,
    ! [A: real] : ~ aa_set_real_o(member_real(A),bot_bot_set_real) ).

% emptyE
tff(fact_1075_emptyE,axiom,
    ! [A: $o] : ~ aa_set_o_o(member_o((A)),bot_bot_set_o) ).

% emptyE
tff(fact_1076_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).

% emptyE
tff(fact_1077_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_1078_equals0D,axiom,
    ! [A2: set_complex,A: complex] :
      ( ( A2 = bot_bot_set_complex )
     => ~ aa_set_complex_o(member_complex(A),A2) ) ).

% equals0D
tff(fact_1079_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ aa_set_set_nat_o(member_set_nat(A),A2) ) ).

% equals0D
tff(fact_1080_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ aa_set_real_o(member_real(A),A2) ) ).

% equals0D
tff(fact_1081_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A)),A2) ) ).

% equals0D
tff(fact_1082_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A),A2) ) ).

% equals0D
tff(fact_1083_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A2) ) ).

% equals0D
tff(fact_1084_equals0I,axiom,
    ! [A2: set_complex] :
      ( ! [Y3: complex] : ~ aa_set_complex_o(member_complex(Y3),A2)
     => ( A2 = bot_bot_set_complex ) ) ).

% equals0I
tff(fact_1085_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(Y3),A2)
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1086_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y3: real] : ~ aa_set_real_o(member_real(Y3),A2)
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1087_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y3: $o] : ~ aa_set_o_o(member_o((Y3)),A2)
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_1088_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] : ~ aa_set_nat_o(member_nat(Y3),A2)
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1089_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y3: int] : ~ aa_set_int_o(member_int(Y3),A2)
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1090_ex__in__conv,axiom,
    ! [A2: set_complex] :
      ( ? [X4: complex] : aa_set_complex_o(member_complex(X4),A2)
    <=> ( A2 != bot_bot_set_complex ) ) ).

% ex_in_conv
tff(fact_1091_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ? [X4: set_nat] : aa_set_set_nat_o(member_set_nat(X4),A2)
    <=> ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1092_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ? [X4: real] : aa_set_real_o(member_real(X4),A2)
    <=> ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1093_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ? [X4: $o] : aa_set_o_o(member_o((X4)),A2)
    <=> ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_1094_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ? [X4: nat] : aa_set_nat_o(member_nat(X4),A2)
    <=> ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1095_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ? [X4: int] : aa_set_int_o(member_int(X4),A2)
    <=> ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1096_length__pos__if__in__set,axiom,
    ! [Xa: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xa),set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1097_length__pos__if__in__set,axiom,
    ! [Xa: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xa),set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1098_length__pos__if__in__set,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1099_length__pos__if__in__set,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xa,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1100_length__pos__if__in__set,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xa)),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1101_length__pos__if__in__set,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xa),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1102_length__pos__if__in__set,axiom,
    ! [Xa: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xa),set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1103_nat__mult__div__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1104_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Q2)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_1105_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1106_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1107_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1108_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1109_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_1110_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_1111_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_1112_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_1113_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_1114_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_1115_dividend__less__times__div,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)))) ) ).

% dividend_less_times_div
tff(fact_1116_dividend__less__div__times,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_1117_split__div,axiom,
    ! [P: fun_nat_o,Ma: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I: nat,J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => ( ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I)),J2) )
               => aa_nat_o(P,I) ) ) ) ) ) ).

% split_div
tff(fact_1118_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_1119_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_1120_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_1121_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_1122_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_1123_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_1124_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_1125_exp__add__not__zero__imp__right,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1126_exp__add__not__zero__imp__right,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1127_exp__add__not__zero__imp__left,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1128_exp__add__not__zero__imp__left,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1129_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_1130_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Ma: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_1131_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Nb: nat,Ma: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_1132_power__odd__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1133_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1134_power__odd__eq,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1135_power__odd__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1136_power__odd__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1137_sum__power2__gt__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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)))))
    <=> ( ( Xa != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1138_sum__power2__gt__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xa),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)))))
    <=> ( ( Xa != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1139_sum__power2__gt__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xa),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)))))
    <=> ( ( Xa != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1140_not__sum__power2__lt__zero,axiom,
    ! [Xa: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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_1141_not__sum__power2__lt__zero,axiom,
    ! [Xa: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xa),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_1142_not__sum__power2__lt__zero,axiom,
    ! [Xa: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xa),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_1143_times__divide__times__eq,axiom,
    ! [Xa: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xa),Z)),aa_complex_complex(times_times_complex(Y),W)) ).

% times_divide_times_eq
tff(fact_1144_times__divide__times__eq,axiom,
    ! [Xa: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_1145_times__divide__times__eq,axiom,
    ! [Xa: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(Xa),Y)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xa),Z)),aa_rat_rat(times_times_rat(Y),W)) ).

% times_divide_times_eq
tff(fact_1146_divide__divide__times__eq,axiom,
    ! [Xa: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(Xa),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xa),W)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_1147_divide__divide__times__eq,axiom,
    ! [Xa: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_1148_divide__divide__times__eq,axiom,
    ! [Xa: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Xa),Y)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xa),W)),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_1149_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_1150_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_1151_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_1152_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_1153_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_1154_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% add_divide_distrib
tff(fact_1155_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xa: nat,Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1156_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xa: nat,Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1157_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1158_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1159_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1160_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1161_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_1162_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1163_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1164_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_1165_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1166_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_1167_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1168_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1169_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_1170_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1171_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1172_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1173_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1174_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xa)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xa)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xa) ) ) ) ).

% member_valid_both_member_options
tff(fact_1175_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_1176_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_1177_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_1178_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_1179_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_1180_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_1181_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_1182_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_1183_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_1184_double__not__eq__Suc__double,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_1185_Suc__double__not__eq__double,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_1186_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xa: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),Xa) ).

% buildup_nothing_in_min_max
tff(fact_1187_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),Xa)
        | aa_nat_o(vEBT_VEBT_membermima(Ta),Xa) ) ) ).

% both_member_options_def
tff(fact_1188_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_1189_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_1190_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1191_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1192_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1193_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1194_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(times_times_int(A),C) = aa_int_int(times_times_int(B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1195_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1196_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1197_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1198_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1199_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(times_times_int(C),A) = aa_int_int(times_times_int(C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1200_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_1201_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_1202_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1203_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_1204_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1205_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1206_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_1207_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1208_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_1209_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_1210_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_1211_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_1212_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1213_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_1214_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(times_times_int(zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_1215_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1216_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1217_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1218_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_1219_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1220_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1221_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_1222_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_1223_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_1224_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1225_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_1226_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_1227_psubsetD,axiom,
    ! [A2: set_complex,B2: set_complex,C: complex] :
      ( aa_set_complex_o(ord_less_set_complex(A2),B2)
     => ( aa_set_complex_o(member_complex(C),A2)
       => aa_set_complex_o(member_complex(C),B2) ) ) ).

% psubsetD
tff(fact_1228_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
     => ( aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% psubsetD
tff(fact_1229_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B2)
     => ( aa_set_set_nat_o(member_set_nat(C),A2)
       => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% psubsetD
tff(fact_1230_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% psubsetD
tff(fact_1231_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => ( aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% psubsetD
tff(fact_1232_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Nb),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_1233_enat__0__less__mult__iff,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(Ma,Nb))
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Ma)
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_1234_linorder__neqE__linordered__idom,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1235_linorder__neqE__linordered__idom,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1236_linorder__neqE__linordered__idom,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1237_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1238_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1239_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1240_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1241_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1242_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(times_times_int(A),C) = aa_int_int(times_times_int(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1243_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1244_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1245_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1246_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1247_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(times_times_int(C),A) = aa_int_int(times_times_int(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1248_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_1249_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_1250_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1251_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_1252_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(times_times_int(A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_1253_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_1254_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_1255_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1256_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_1257_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_1258_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_1259_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_1260_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1261_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_1262_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_1263_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1264_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1265_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1266_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E2)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E2)),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1267_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_1268_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_1269_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_1270_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% distrib_right
tff(fact_1271_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_1272_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_1273_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_1274_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% distrib_left
tff(fact_1275_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1276_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1277_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1278_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1279_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1280_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1281_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1282_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1283_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1284_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1285_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1286_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1287_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1288_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1289_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1290_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1291_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1292_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1293_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1294_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1295_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1296_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1297_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1298_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1299_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1300_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1301_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1302_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1303_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1304_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1305_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1306_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1307_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1308_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1309_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1310_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1311_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1312_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1313_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1314_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1315_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1316_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1317_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1318_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1319_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_1320_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1321_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1322_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_1323_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_1324_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_1325_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_1326_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_1327_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_1328_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_1329_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_1330_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1331_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1332_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1333_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1334_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_1335_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_1336_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_1337_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_1338_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_1339_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_1340_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_1341_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_1342_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1343_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1344_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1345_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_1346_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_1347_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_1348_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1349_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1350_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1351_add__less__zeroD,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1352_add__less__zeroD,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xa),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1353_add__less__zeroD,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Xa),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1354_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1355_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1356_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1357_cong__exp__iff__simps_I9_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1358_cong__exp__iff__simps_I9_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1359_cong__exp__iff__simps_I9_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1360_cong__exp__iff__simps_I4_J,axiom,
    ! [Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1361_cong__exp__iff__simps_I4_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1362_cong__exp__iff__simps_I4_J,axiom,
    ! [Ma: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_1363_mod__eq__0D,axiom,
    ! [Ma: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),D) = zero_zero_nat )
     => ? [Q3: nat] : Ma = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_1364_nat__mod__eq__iff,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1365_not__sum__squares__lt__zero,axiom,
    ! [Xa: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_1366_not__sum__squares__lt__zero,axiom,
    ! [Xa: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xa),Xa)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_1367_not__sum__squares__lt__zero,axiom,
    ! [Xa: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xa),Xa)),aa_int_int(times_times_int(Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_1368_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1369_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1370_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1371_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_1372_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_1373_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_1374_cong__exp__iff__simps_I8_J,axiom,
    ! [Ma: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),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_1375_cong__exp__iff__simps_I8_J,axiom,
    ! [Ma: num,Q2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_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_1376_cong__exp__iff__simps_I8_J,axiom,
    ! [Ma: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1377_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1378_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1379_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1380_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1381_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(plus_plus_int(A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1382_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_1383_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1384_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(plus_plus_int(A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1385_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_1386_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_1387_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_decomp
tff(fact_1388_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_1389_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_1390_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).

% div_mult_mod_eq
tff(fact_1391_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_1392_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_1393_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),B)),aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1394_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1395_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_1396_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),B)),aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B))) = A ).

% mod_mult_div_eq
tff(fact_1397_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_1398_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_1399_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).

% mult_div_mod_eq
tff(fact_1400_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_1401_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_1402_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1403_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1404_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(plus_plus_int(A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1405_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_1406_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_1407_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,Sb: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,Sb),Xa) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,Sb) ).

% vebt_insert.simps(3)
tff(fact_1408_set__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1409_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1410_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1411_flip__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1412_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1413_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1414_unset__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1415_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1416_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1417_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,Sb: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,Sb),Xa) = vEBT_Node(Info,zero_zero_nat,Ts,Sb) ).

% vebt_insert.simps(2)
tff(fact_1418_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Ma: nat,Nb: nat] :
      ( ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M2),zero_zero_nat)
     => ( ! [M2: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M2),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Ma),Nb) ) ) ).

% gcd_nat_induct
tff(fact_1419_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_1420_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X22)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_1421_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_1422_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_1423_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_1424_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => ( ( aa_int_int(modulo_modulo_int(A2),Nb) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B2),Nb) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A2),Nb)),aa_int_int(divide_divide_int(B2),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1425_div__mod__decomp__int,axiom,
    ! [A2: int,Nb: int] : A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(divide_divide_int(A2),Nb)),Nb)),aa_int_int(modulo_modulo_int(A2),Nb)) ).

% div_mod_decomp_int
tff(fact_1426_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_1427_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1428_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1429_zmod__eq__0__iff,axiom,
    ! [Ma: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(Ma),D) = zero_zero_int )
    <=> ? [Q4: int] : Ma = aa_int_int(times_times_int(D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_1430_zmod__eq__0D,axiom,
    ! [Ma: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(Ma),D) = zero_zero_int )
     => ? [Q3: int] : Ma = aa_int_int(times_times_int(D),Q3) ) ).

% zmod_eq_0D
tff(fact_1431_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(times_times_int(K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_1432_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(times_times_int(zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_1433_imult__is__0,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(Ma,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Ma = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_1434_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1435_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L)) ) ).

% neg_mod_bound
tff(fact_1436_zmult__zless__mono2,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(K),Ia)),aa_int_int(times_times_int(K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1437_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_1438_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(plus_plus_int(zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_1439_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(plus_plus_int(K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_1440_iadd__is__0,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Ma,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Ma = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_1441_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z2: int] : aa_int_int(times_times_int(W),aa_int_int(plus_plus_int(Z1),Z2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Z1)),aa_int_int(times_times_int(W),Z2)) ).

% int_distrib(2)
tff(fact_1442_int__distrib_I1_J,axiom,
    ! [Z1: int,Z2: int,W: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(Z1),Z2)),W) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Z1),W)),aa_int_int(times_times_int(Z2),W)) ).

% int_distrib(1)
tff(fact_1443_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_1444_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X22)) ).

% VEBT.distinct(1)
tff(fact_1445_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X222: $o] : Y != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_1446_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1447_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1448_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1449_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1450_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_1451_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_1452_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_1453_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( ( Xa != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1454_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_1455_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_1456_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),Nb) = A )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),Nb) = A ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1457_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B4)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B4),A4) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),zero_zero_nat)
       => ( ! [A4: nat,B4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B4)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_1458_four__x__squared,axiom,
    ! [Xa: 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,Xa),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))),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_1459_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] : vEBT_VEBT_insert(vEBT_Leaf((A),(B)),Xa) = vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xa) ).

% VEBT_internal.insert'.simps(1)
tff(fact_1460_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1461_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1462_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1463_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_1464_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_1465_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Nb),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(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_1466_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_1467_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_1468_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1469_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1470_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1471_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1472_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(R2),C)) != aa_int_int(plus_plus_int(B),aa_int_int(times_times_int(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1473_mult__less__iff1,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_1474_mult__less__iff1,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Xa),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_1475_mult__less__iff1,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(Xa),Z)),aa_int_int(times_times_int(Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_1476_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_1477_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_1478_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Nb),K),L) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_concat_bit(Nb,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_1479_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ) ).

% deg_1_Leafy
tff(fact_1480_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ).

% deg_1_Leaf
tff(fact_1481_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).

% deg1Leaf
tff(fact_1482_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_1483_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_1484_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_1485_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_1486_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_1487_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_1488_nat__1__eq__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) )
    <=> ( ( Ma = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1489_nat__mult__eq__1__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) = one_one_nat )
    <=> ( ( Ma = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1490_real__divide__square__eq,axiom,
    ! [R2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R2) ).

% real_divide_square_eq
tff(fact_1491_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_1492_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_1493_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_1494_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_1495_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),one_one_int) = A ).

% mult.right_neutral
tff(fact_1496_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_1497_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_1498_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_1499_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_1500_mult__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(one_one_int),A) = A ).

% mult_1
tff(fact_1501_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_1502_div__by__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).

% div_by_1
tff(fact_1503_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_1504_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_1505_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_1506_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_1507_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% bits_div_by_1
tff(fact_1508_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_1509_power__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).

% power_one
tff(fact_1510_power__one,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Nb) = one_one_real ).

% power_one
tff(fact_1511_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_1512_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_1513_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_1514_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_1515_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_1516_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_1517_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1518_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1519_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(times_times_int(C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1520_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_1521_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1522_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1523_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(times_times_int(C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1524_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_1525_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1526_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1527_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(times_times_int(B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1528_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_1529_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_1530_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1531_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(times_times_int(A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1532_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_1533_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_1534_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_1535_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_1536_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_1537_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_1538_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_1539_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_1540_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_1541_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_1542_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_1543_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_1544_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_1545_divide__self__if,axiom,
    ! [A: complex] :
      aa_complex_complex(divide1717551699836669952omplex(A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_1546_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_1547_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(divide_divide_rat(A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_1548_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_1549_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_1550_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_1551_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_1552_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1553_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1554_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_1555_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1556_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1557_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_Extended_enat(numera1916890842035813515d_enat,Nb) = one_on7984719198319812577d_enat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_1558_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_1559_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_1560_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_1561_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_1562_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_1563_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_on7984719198319812577d_enat = aa_num_Extended_enat(numera1916890842035813515d_enat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_1564_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_1565_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_1566_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_1567_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_1568_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_1569_power__inject__exp,axiom,
    ! [A: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% power_inject_exp
tff(fact_1570_power__inject__exp,axiom,
    ! [A: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),Ma) = aa_nat_rat(power_power_rat(A),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% power_inject_exp
tff(fact_1571_power__inject__exp,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% power_inject_exp
tff(fact_1572_power__inject__exp,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),Ma) = aa_nat_int(power_power_int(A),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% power_inject_exp
tff(fact_1573_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_1574_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_1575_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_1576_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_1577_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_1578_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_1579_not__real__square__gt__zero,axiom,
    ! [Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa))
    <=> ( Xa = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_1580_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_1581_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_1582_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Nb,K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_1583_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_1584_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_1585_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_1586_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_1587_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_1588_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_1589_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_1590_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_1591_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_1592_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_1593_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_1594_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_1595_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_1596_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1597_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1598_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_1599_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1600_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1601_power__strict__increasing__iff,axiom,
    ! [B: real,Xa: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_1602_power__strict__increasing__iff,axiom,
    ! [B: rat,Xa: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Xa)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_1603_power__strict__increasing__iff,axiom,
    ! [B: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_1604_power__strict__increasing__iff,axiom,
    ! [B: int,Xa: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Xa)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_1605_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_1606_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),Nb))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1607_one__add__one,axiom,
    plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,one_on7984719198319812577d_enat) = aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1608_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_1609_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1610_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_1611_one__add__one,axiom,
    aa_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_1612_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1613_power__strict__decreasing__iff,axiom,
    ! [B: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1614_power__strict__decreasing__iff,axiom,
    ! [B: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Ma)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1615_power__strict__decreasing__iff,axiom,
    ! [B: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1616_power__strict__decreasing__iff,axiom,
    ! [B: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Ma)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1617_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_1618_bits__one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_1619_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_1620_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_1621_one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_1622_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_1623_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_1624_numeral__plus__one,axiom,
    ! [Nb: num] : plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Nb),one_on7984719198319812577d_enat) = aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1625_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1626_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1627_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1628_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1629_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1630_one__plus__numeral,axiom,
    ! [Nb: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)) = aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1631_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1632_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1633_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1634_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1635_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1636_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_1637_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_1638_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_1639_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_1640_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_1641_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_1642_bits__1__div__2,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_1643_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_1644_one__div__two__eq__zero,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_1645_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_1646_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1647_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_1648_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_1649_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1650_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_1651_mod2__gr__0,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_1652_real__arch__pow__inv,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1653_real__arch__pow,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),N)) ) ).

% real_arch_pow
tff(fact_1654_one__reorient,axiom,
    ! [Xa: complex] :
      ( ( one_one_complex = Xa )
    <=> ( Xa = one_one_complex ) ) ).

% one_reorient
tff(fact_1655_one__reorient,axiom,
    ! [Xa: real] :
      ( ( one_one_real = Xa )
    <=> ( Xa = one_one_real ) ) ).

% one_reorient
tff(fact_1656_one__reorient,axiom,
    ! [Xa: rat] :
      ( ( one_one_rat = Xa )
    <=> ( Xa = one_one_rat ) ) ).

% one_reorient
tff(fact_1657_one__reorient,axiom,
    ! [Xa: nat] :
      ( ( one_one_nat = Xa )
    <=> ( Xa = one_one_nat ) ) ).

% one_reorient
tff(fact_1658_one__reorient,axiom,
    ! [Xa: int] :
      ( ( one_one_int = Xa )
    <=> ( Xa = one_one_int ) ) ).

% one_reorient
tff(fact_1659_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(plus_plus_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(plus_plus_int(K),L)) ).

% signed_take_bit_add
tff(fact_1660_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(times_times_int(K),L)) ).

% signed_take_bit_mult
tff(fact_1661_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1662_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1663_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1664_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1665_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1666_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_1667_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_1668_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_1669_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_1670_less__numeral__extra_I4_J,axiom,
    ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),one_on7984719198319812577d_enat) ).

% less_numeral_extra(4)
tff(fact_1671_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_1672_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_1673_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1674_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_1675_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1676_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_1677_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_1678_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_1679_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_1680_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_1681_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_1682_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_1683_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_1684_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_1685_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Ma: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(Nb,K),aa_int_int(bit_concat_bit(Ma,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb),aa_int_int(bit_concat_bit(Nb,K),L)),R2) ).

% concat_bit_assoc
tff(fact_1686_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_1687_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_1688_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_1689_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_1690_not__one__less__zero,axiom,
    ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) ).

% not_one_less_zero
tff(fact_1691_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_1692_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_1693_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_1694_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_1695_zero__less__one,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) ).

% zero_less_one
tff(fact_1696_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_1697_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_1698_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_1699_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_1700_less__numeral__extra_I1_J,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) ).

% less_numeral_extra(1)
tff(fact_1701_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)),one_on7984719198319812577d_enat) ).

% not_numeral_less_one
tff(fact_1702_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_1703_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_1704_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_1705_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_1706_less__1__mult,axiom,
    ! [Ma: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ma)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ma),Nb)) ) ) ).

% less_1_mult
tff(fact_1707_less__1__mult,axiom,
    ! [Ma: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Ma)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(Ma),Nb)) ) ) ).

% less_1_mult
tff(fact_1708_less__1__mult,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ) ) ).

% less_1_mult
tff(fact_1709_less__1__mult,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(times_times_int(Ma),Nb)) ) ) ).

% less_1_mult
tff(fact_1710_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_1711_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).

% less_add_one
tff(fact_1712_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_1713_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(plus_plus_int(A),one_one_int)) ).

% less_add_one
tff(fact_1714_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_1715_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_1716_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_1717_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),one_one_int)),aa_int_int(plus_plus_int(B),one_one_int)) ) ).

% add_mono1
tff(fact_1718_add__mono1,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,plus_p3455044024723400733d_enat(A,one_on7984719198319812577d_enat)),plus_p3455044024723400733d_enat(B,one_on7984719198319812577d_enat)) ) ).

% add_mono1
tff(fact_1719_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_1720_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_1721_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_1722_one__plus__numeral__commute,axiom,
    ! [Xa: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Xa)) = plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Xa),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_1723_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xa)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xa)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_1724_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xa)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_1725_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_1726_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,Xa)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Xa)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_1727_one__plus__numeral__commute,axiom,
    ! [Xa: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xa)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xa)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_1728_numeral__One,axiom,
    aa_num_Extended_enat(numera1916890842035813515d_enat,one) = one_on7984719198319812577d_enat ).

% numeral_One
tff(fact_1729_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_1730_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_1731_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_1732_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_1733_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_1734_left__right__inverse__power,axiom,
    ! [Xa: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xa),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xa),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_1735_left__right__inverse__power,axiom,
    ! [Xa: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_1736_left__right__inverse__power,axiom,
    ! [Xa: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xa),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xa),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_1737_left__right__inverse__power,axiom,
    ! [Xa: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),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,Xa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_1738_left__right__inverse__power,axiom,
    ! [Xa: int,Y: int,Nb: nat] :
      ( ( aa_int_int(times_times_int(Xa),Y) = one_one_int )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(Xa),Nb)),aa_nat_int(power_power_int(Y),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_1739_power__one__over,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),Nb) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_one_over
tff(fact_1740_power__one__over,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_one_over
tff(fact_1741_power__one__over,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),Nb) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_one_over
tff(fact_1742_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_1743_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_1744_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_1745_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_1746_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_1747_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_1748_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1749_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1750_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1751_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_1752_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1753_mult__eq__self__implies__10,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Ma = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1754_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1755_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_1756_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1757_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_1758_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_1759_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_1760_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_1761_zero__less__two,axiom,
    aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,one_on7984719198319812577d_enat)) ).

% zero_less_two
tff(fact_1762_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_1763_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_1764_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_1765_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_1766_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_1767_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(B),A)),B) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_1768_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_1769_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),B) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_1770_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_1771_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_1772_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_1773_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_1774_power__gt1__lemma,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_1775_power__gt1__lemma,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_1776_power__gt1__lemma,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_1777_power__gt1__lemma,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_1778_power__less__power__Suc,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_1779_power__less__power__Suc,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_1780_power__less__power__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_1781_power__less__power__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_1782_power__gt1,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1783_power__gt1,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1784_power__gt1,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1785_power__gt1,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1786_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_1787_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_1788_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_1789_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_1790_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_1791_power__less__imp__less__exp,axiom,
    ! [A: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1792_power__less__imp__less__exp,axiom,
    ! [A: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Ma)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1793_power__less__imp__less__exp,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1794_power__less__imp__less__exp,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Ma)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1795_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)) ) ) ).

% power_strict_increasing
tff(fact_1796_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N3)) ) ) ).

% power_strict_increasing
tff(fact_1797_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)) ) ) ).

% power_strict_increasing
tff(fact_1798_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N3)) ) ) ).

% power_strict_increasing
tff(fact_1799_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_1800_div__eq__dividend__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = Ma )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_1801_div__less__dividend,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Ma) ) ) ).

% div_less_dividend
tff(fact_1802_pos__zmult__eq__1__iff,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ma)
     => ( ( aa_int_int(times_times_int(Ma),Nb) = one_one_int )
      <=> ( ( Ma = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1803_int__div__less__self,axiom,
    ! [Xa: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(Xa),K)),Xa) ) ) ).

% int_div_less_self
tff(fact_1804_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),Xa)
    <=> $ite(
          Xa = zero_zero_nat,
          (A),
          $ite(Xa = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_1805_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xa) = $ite(
        Xa = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xa = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_1806_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),Xa)
    <=> $ite(
          Xa = zero_zero_nat,
          (A),
          $ite(Xa = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1807_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_1808_power__Suc__less,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_1809_power__Suc__less,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_1810_power__Suc__less,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_1811_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_1812_power__Suc__less__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_1813_power__Suc__less__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_1814_power__Suc__less__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_1815_power__Suc__less__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_1816_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1817_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1818_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1819_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1820_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_1821_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_1822_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_1823_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_1824_one__power2,axiom,
    aa_nat_complex(power_power_complex(one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_1825_one__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% one_less_power
tff(fact_1826_one__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% one_less_power
tff(fact_1827_one__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% one_less_power
tff(fact_1828_one__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% one_less_power
tff(fact_1829_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_1830_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_1831_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_1832_Suc__times__mod__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
     => ( 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,Ma),Nb))),Ma) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_1833_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_1834_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_1835_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_1836_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_1837_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(plus_plus_int(B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_1838_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1839_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1840_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1841_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) != aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1842_crossproduct__eq,axiom,
    ! [W: real,Y: real,Xa: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)) )
    <=> ( ( W = Xa )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_1843_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,Xa: rat,Z: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Xa),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Z)),aa_rat_rat(times_times_rat(Xa),Y)) )
    <=> ( ( W = Xa )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_1844_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,Xa: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y)) )
    <=> ( ( W = Xa )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_1845_crossproduct__eq,axiom,
    ! [W: int,Y: int,Xa: int,Z: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Y)),aa_int_int(times_times_int(Xa),Z)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Z)),aa_int_int(times_times_int(Xa),Y)) )
    <=> ( ( W = Xa )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_1846_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_1847_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Ma: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( some_nat(Ma) = vEBT_vebt_mint(Ta) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% misiz
tff(fact_1848_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1849_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_1850_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_1851_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1852_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1853_even__succ__mod__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1854_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1855_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_1856_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_1857_even__succ__div__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_1858_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(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_1859_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1860_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1861_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(Xa,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != vEBT_vebt_insert(vEBT_Leaf((A4),(B4)),Xaa) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,Deg,TreeList2,Summary2) )
             => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)),Xaa),vEBT_Node(Info2,Deg,TreeList2,Summary2),vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList2,Summary2),Xaa)) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_1862_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_1863_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_1864_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_1865_arith__geo__mean,axiom,
    ! [U: real,Xa: 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,Xa),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_1866_arith__geo__mean,axiom,
    ! [U: rat,Xa: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(Xa),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xa),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_1867_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_1868_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_maxt(Ta) = some_nat(Xa) )
     => aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xa) ) ).

% maxbmo
tff(fact_1869_add__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_1870_mul__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_1871_max__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xa)
    <=> ( aa_set_nat_o(member_nat(Xa),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa) ) ) ) ).

% max_in_set_def
tff(fact_1872_min__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xa)
    <=> ( aa_set_nat_o(member_nat(Xa),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4) ) ) ) ).

% min_in_set_def
tff(fact_1873_power__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_1874_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_1875_semiring__norm_I71_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% semiring_norm(71)
tff(fact_1876_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_1877_subset__empty,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real)
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1878_subset__empty,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o)
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_1879_subset__empty,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1880_subset__empty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1881_empty__subsetI,axiom,
    ! [A2: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A2) ).

% empty_subsetI
tff(fact_1882_empty__subsetI,axiom,
    ! [A2: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A2) ).

% empty_subsetI
tff(fact_1883_empty__subsetI,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A2) ).

% empty_subsetI
tff(fact_1884_empty__subsetI,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A2) ).

% empty_subsetI
tff(fact_1885_nat__dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),one_one_nat)
    <=> ( Ma = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_1886_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( ( A2 != B2 )
       => aa_set_nat_o(ord_less_set_nat(A2),B2) ) ) ).

% psubsetI
tff(fact_1887_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Mini) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),Xa)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xa) ) ) ) ).

% mint_corr_help
tff(fact_1888_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),Xa)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_1889_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_1890_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Ma)),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_1891_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_1892_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_1893_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_1894_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_1895_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_1896_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_1897_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_1898_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_1899_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_1900_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_1901_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_1902_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_1903_semiring__norm_I69_J,axiom,
    ! [Ma: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Ma)),one) ).

% semiring_norm(69)
tff(fact_1904_dvd__0__right,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_1905_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),zero_zero_complex) ).

% dvd_0_right
tff(fact_1906_dvd__0__right,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),zero_zero_real) ).

% dvd_0_right
tff(fact_1907_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_1908_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_1909_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_1910_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_1911_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_1912_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_1913_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_1914_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_1915_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_1916_dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Ma = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_1917_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_1918_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1919_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1920_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1921_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1922_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_1923_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1924_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1925_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1926_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1927_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_1928_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(B,A)),divide6298287555418463151nteger(C,A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_1929_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_1930_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(B),A)),aa_int_int(divide_divide_int(C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_1931_Suc__le__mono,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Ma))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ).

% Suc_le_mono
tff(fact_1932_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_1933_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_1934_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1935_nat__add__left__cancel__le,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_1936_not__None__eq,axiom,
    ! [Xa: option_nat] :
      ( ( Xa != none_nat )
    <=> ? [Y5: nat] : Xa = some_nat(Y5) ) ).

% not_None_eq
tff(fact_1937_not__None__eq,axiom,
    ! [Xa: option4927543243414619207at_nat] :
      ( ( Xa != none_P5556105721700978146at_nat )
    <=> ? [Y5: product_prod_nat_nat] : Xa = some_P7363390416028606310at_nat(Y5) ) ).

% not_None_eq
tff(fact_1938_not__None__eq,axiom,
    ! [Xa: option_num] :
      ( ( Xa != none_num )
    <=> ? [Y5: num] : Xa = some_num(Y5) ) ).

% not_None_eq
tff(fact_1939_not__Some__eq,axiom,
    ! [Xa: option_nat] :
      ( ! [Y5: nat] : Xa != some_nat(Y5)
    <=> ( Xa = none_nat ) ) ).

% not_Some_eq
tff(fact_1940_not__Some__eq,axiom,
    ! [Xa: option4927543243414619207at_nat] :
      ( ! [Y5: product_prod_nat_nat] : Xa != some_P7363390416028606310at_nat(Y5)
    <=> ( Xa = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_1941_not__Some__eq,axiom,
    ! [Xa: option_num] :
      ( ! [Y5: num] : Xa != some_num(Y5)
    <=> ( Xa = none_num ) ) ).

% not_Some_eq
tff(fact_1942_of__bool__eq_I1_J,axiom,
    aa_o_complex(zero_n1201886186963655149omplex,$false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_1943_of__bool__eq_I1_J,axiom,
    aa_o_real(zero_n3304061248610475627l_real,$false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_1944_of__bool__eq_I1_J,axiom,
    aa_o_rat(zero_n2052037380579107095ol_rat,$false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_1945_of__bool__eq_I1_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_1946_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_1947_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger($false) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_1948_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_complex(zero_n1201886186963655149omplex,(P)) = zero_zero_complex )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_1949_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_real(zero_n3304061248610475627l_real,(P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_1950_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_rat(zero_n2052037380579107095ol_rat,(P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_1951_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_1952_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = zero_zero_int )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_1953_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = zero_z3403309356797280102nteger )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_1954_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(zero_n3304061248610475627l_real,(P))),aa_o_real(zero_n3304061248610475627l_real,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1955_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),aa_o_rat(zero_n2052037380579107095ol_rat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1956_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_n1046097342994218471d_enat((P))),zero_n1046097342994218471d_enat((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1957_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1958_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1959_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_1960_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(Nb,K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_1961_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1962_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_1963_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_1964_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_1965_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_1966_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1967_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_1968_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1969_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1970_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1971_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1972_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1973_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1974_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_1975_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_1976_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_1977_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(plus_plus_int(B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_1978_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_1979_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_1980_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_1981_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(plus_plus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_1982_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_1983_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_1984_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_1985_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_1986_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_1987_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_1988_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_1989_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_1990_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1991_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1992_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(B),A)),aa_int_int(times_times_int(C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1993_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1994_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1995_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1996_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_1997_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_1998_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_1999_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_2000_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_2001_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_2002_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_2003_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_2004_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_2005_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_2006_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2007_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2008_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2009_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2010_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2011_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2012_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2013_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2014_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2015_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),aa_int_int(times_times_int(C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2016_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_2017_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_2018_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_2019_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_2020_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_2021_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(times_times_int(A),aa_int_int(divide_divide_int(B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_2022_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_2023_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_2024_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_2025_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_add
tff(fact_2026_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_2027_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_add
tff(fact_2028_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_2029_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_2030_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_2031_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_2032_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_2033_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_2034_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_2035_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_2036_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(divide_divide_int(one_one_int),aa_int_int(divide_divide_int(one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_2037_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_2038_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_2039_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_2040_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(zero_n3304061248610475627l_real,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_2041_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_2042_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_2043_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_2044_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_2045_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(zero_n3304061248610475627l_real,(P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_2046_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_2047_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_2048_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_2049_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_2050_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_2051_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2052_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2053_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2054_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2055_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xa)
       => ( vEBT_vebt_mint(Ta) = some_nat(Xa) ) ) ) ).

% mint_sound
tff(fact_2056_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Xa) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xa) ) ) ).

% mint_corr
tff(fact_2057_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xa)
       => ( vEBT_vebt_maxt(Ta) = some_nat(Xa) ) ) ) ).

% maxt_sound
tff(fact_2058_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Xa) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xa) ) ) ).

% maxt_corr
tff(fact_2059_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_2060_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_2061_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_2062_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_2063_enat__ord__number_I1_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Ma)),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_2064_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2065_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2066_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_2067_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_2068_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)),one_on7984719198319812577d_enat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2069_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2070_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2071_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2072_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2073_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2074_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2075_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2076_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2077_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_2078_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_2079_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_2080_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_2081_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_2082_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_2083_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_2084_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(one_one_int),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% unit_mult_div_div
tff(fact_2085_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_2086_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_2087_one__le__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_2088_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2089_mult__le__cancel2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% mult_le_cancel2
tff(fact_2090_lesseq__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(Xa)),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_2091_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2092_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2093_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2094_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2095_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2096_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2097_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2098_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2099_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_2100_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_2101_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(times_times_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_2102_power__increasing__iff,axiom,
    ! [B: real,Xa: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2103_power__increasing__iff,axiom,
    ! [B: rat,Xa: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xa)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2104_power__increasing__iff,axiom,
    ! [B: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2105_power__increasing__iff,axiom,
    ! [B: int,Xa: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Xa)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_2106_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2107_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2108_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2109_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2110_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_2111_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_2112_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_2113_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_2114_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_2115_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_2116_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_2117_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_2118_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_2119_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_2120_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_2121_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_2122_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_2123_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_2124_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_2125_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xa),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        & ! [Z3: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z3) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3) ) ) ) ).

% succ_member
tff(fact_2126_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xa),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        & ! [Z3: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),Xa) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),Y) ) ) ) ).

% pred_member
tff(fact_2127_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2128_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2129_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2130_power2__eq__iff__nonneg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2131_power2__eq__iff__nonneg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(Xa),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))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2132_power2__eq__iff__nonneg,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),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))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2133_power2__eq__iff__nonneg,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(power_power_int(Xa),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))) )
        <=> ( Xa = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2134_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_2135_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_2136_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_2137_power__decreasing__iff,axiom,
    ! [B: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ) ) ).

% power_decreasing_iff
tff(fact_2138_power__decreasing__iff,axiom,
    ! [B: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Ma)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ) ) ).

% power_decreasing_iff
tff(fact_2139_power__decreasing__iff,axiom,
    ! [B: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ) ) ).

% power_decreasing_iff
tff(fact_2140_power__decreasing__iff,axiom,
    ! [B: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Ma)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ) ) ).

% power_decreasing_iff
tff(fact_2141_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2142_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2143_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2144_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_2145_power__less__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_2146_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_2147_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_2148_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_2149_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_2150_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_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_2151_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(divide_divide_int(aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_2152_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_2153_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2154_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2155_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2156_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_2157_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_2158_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_2159_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_2160_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_2161_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_2162_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_2163_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_2164_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_2165_even__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_2166_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_2167_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_2168_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_2169_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_2170_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_2171_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2172_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2173_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2174_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_2175_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_2176_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_2177_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_2178_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_2179_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_2180_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_2181_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_2182_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,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_2183_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(zero_n3304061248610475627l_real,(P))) ).

% zero_less_eq_of_bool
tff(fact_2184_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(P))) ).

% zero_less_eq_of_bool
tff(fact_2185_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P))) ).

% zero_less_eq_of_bool
tff(fact_2186_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% zero_less_eq_of_bool
tff(fact_2187_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P))) ).

% zero_less_eq_of_bool
tff(fact_2188_dvd__power__le,axiom,
    ! [Xa: code_integer,Y: code_integer,Nb: nat,Ma: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Ma)) ) ) ).

% dvd_power_le
tff(fact_2189_dvd__power__le,axiom,
    ! [Xa: nat,Y: nat,Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Ma)) ) ) ).

% dvd_power_le
tff(fact_2190_dvd__power__le,axiom,
    ! [Xa: real,Y: real,Nb: nat,Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Ma)) ) ) ).

% dvd_power_le
tff(fact_2191_dvd__power__le,axiom,
    ! [Xa: int,Y: int,Nb: nat,Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xa),Nb)),aa_nat_int(power_power_int(Y),Ma)) ) ) ).

% dvd_power_le
tff(fact_2192_dvd__power__le,axiom,
    ! [Xa: complex,Y: complex,Nb: nat,Ma: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xa),Nb)),aa_nat_complex(power_power_complex(Y),Ma)) ) ) ).

% dvd_power_le
tff(fact_2193_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,Ma: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_2194_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_2195_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_2196_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_2197_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,Ma: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_2198_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Ma)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2199_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2200_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2201_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Ma)),aa_nat_int(power_power_int(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2202_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Ma)),aa_nat_complex(power_power_complex(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2203_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_2204_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_2205_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_2206_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_2207_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% verit_comp_simplify1(2)
tff(fact_2208_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2209_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2210_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2211_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2212_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => ? [X3: nat] :
            ( aa_nat_o(P,X3)
            & ! [Y4: nat] :
                ( aa_nat_o(P,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_2213_nat__le__linear,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ).

% nat_le_linear
tff(fact_2214_le__antisym,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( Ma = Nb ) ) ) ).

% le_antisym
tff(fact_2215_eq__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% eq_imp_le
tff(fact_2216_le__trans,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),K) ) ) ).

% le_trans
tff(fact_2217_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_2218_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2219_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2220_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2221_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2222_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2223_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,Nb)),aa_nat_set_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_2224_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_2225_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_2226_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_2227_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_2228_zdvd__antisym__nonneg,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ma),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Ma)
           => ( Ma = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2229_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% dvd_imp_le
tff(fact_2230_power__increasing,axiom,
    ! [Nb: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)) ) ) ).

% power_increasing
tff(fact_2231_power__increasing,axiom,
    ! [Nb: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N3)) ) ) ).

% power_increasing
tff(fact_2232_power__increasing,axiom,
    ! [Nb: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)) ) ) ).

% power_increasing
tff(fact_2233_power__increasing,axiom,
    ! [Nb: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N3)) ) ) ).

% power_increasing
tff(fact_2234_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_2235_power__dvd__imp__le,axiom,
    ! [Ia: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Ia)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_2236_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_real(zero_n3304061248610475627l_real,
        ( (P)
        & (Q) )) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_o_real(zero_n3304061248610475627l_real,(P))),aa_o_real(zero_n3304061248610475627l_real,(Q))) ).

% of_bool_conj
tff(fact_2237_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_rat(zero_n2052037380579107095ol_rat,
        ( (P)
        & (Q) )) = aa_rat_rat(times_times_rat(aa_o_rat(zero_n2052037380579107095ol_rat,(P))),aa_o_rat(zero_n2052037380579107095ol_rat,(Q))) ).

% of_bool_conj
tff(fact_2238_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_conj
tff(fact_2239_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        & (Q) )) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_conj
tff(fact_2240_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        & (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q))) ).

% of_bool_conj
tff(fact_2241_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B)
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_2242_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_2243_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_2244_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_2245_dvd__0__left,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_2246_dvd__0__left,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_2247_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_2248_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_2249_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_2250_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != some_nat(X2) ).

% option.distinct(1)
tff(fact_2251_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_2252_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_2253_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = some_nat(X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_2254_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_2255_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_2256_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != some_nat(X23) ) ).

% option.exhaust
tff(fact_2257_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_2258_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_2259_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_1: option_nat] : aa_option_nat_o(P,X_1)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X4: nat] : aa_option_nat_o(P,some_nat(X4)) ) ) ).

% split_option_ex
tff(fact_2260_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_1: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_1)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_ex
tff(fact_2261_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_1: option_num] : aa_option_num_o(P,X_1)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_ex
tff(fact_2262_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_1: option_nat] : aa_option_nat_o(P,X_1)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X4: nat] : aa_option_nat_o(P,some_nat(X4)) ) ) ).

% split_option_all
tff(fact_2263_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_1: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_1)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_all
tff(fact_2264_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_1: option_num] : aa_option_num_o(P,X_1)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_all
tff(fact_2265_combine__options__cases,axiom,
    ! [Xa: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
      ( ( ( Xa = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xa),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xa),Y) )
       => ( ! [A4: nat,B4: nat] :
              ( ( Xa = some_nat(A4) )
             => ( ( Y = some_nat(B4) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xa),Y) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2266_combine__options__cases,axiom,
    ! [Xa: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xa = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xa),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xa),Y) )
       => ( ! [A4: nat,B4: product_prod_nat_nat] :
              ( ( Xa = some_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xa),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2267_combine__options__cases,axiom,
    ! [Xa: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
      ( ( ( Xa = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xa),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xa),Y) )
       => ( ! [A4: nat,B4: num] :
              ( ( Xa = some_nat(A4) )
             => ( ( Y = some_num(B4) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xa),Y) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2268_combine__options__cases,axiom,
    ! [Xa: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
      ( ( ( Xa = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xa),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xa),Y) )
       => ( ! [A4: product_prod_nat_nat,B4: nat] :
              ( ( Xa = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_nat(B4) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xa),Y) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2269_combine__options__cases,axiom,
    ! [Xa: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xa = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xa),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xa),Y) )
       => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( Xa = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xa),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2270_combine__options__cases,axiom,
    ! [Xa: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( Xa = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xa),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xa),Y) )
       => ( ! [A4: product_prod_nat_nat,B4: num] :
              ( ( Xa = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_num(B4) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xa),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2271_combine__options__cases,axiom,
    ! [Xa: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
      ( ( ( Xa = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xa),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xa),Y) )
       => ( ! [A4: num,B4: nat] :
              ( ( Xa = some_num(A4) )
             => ( ( Y = some_nat(B4) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xa),Y) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2272_combine__options__cases,axiom,
    ! [Xa: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xa = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xa),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xa),Y) )
       => ( ! [A4: num,B4: product_prod_nat_nat] :
              ( ( Xa = some_num(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B4) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xa),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2273_combine__options__cases,axiom,
    ! [Xa: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( Xa = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xa),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xa),Y) )
       => ( ! [A4: num,B4: num] :
              ( ( Xa = some_num(A4) )
             => ( ( Y = some_num(B4) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xa),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xa),Y) ) ) ) ).

% combine_options_cases
tff(fact_2274_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y3) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y3),B) ) ) ) ).

% dvd_productE
tff(fact_2275_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(times_times_int(A),B))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int_int(times_times_int(X3),Y3) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y3),B) ) ) ) ).

% dvd_productE
tff(fact_2276_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B6: nat,C3: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B6),C3) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B6),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C3),C) ) ) ).

% division_decomp
tff(fact_2277_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),C))
     => ? [B6: int,C3: int] :
          ( ( A = aa_int_int(times_times_int(B6),C3) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B6),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C3),C) ) ) ).

% division_decomp
tff(fact_2278_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).

% dvdE
tff(fact_2279_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K2: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvdE
tff(fact_2280_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K2: rat] : A != aa_rat_rat(times_times_rat(B),K2) ) ).

% dvdE
tff(fact_2281_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_2282_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K2: int] : A != aa_int_int(times_times_int(B),K2) ) ).

% dvdE
tff(fact_2283_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_2284_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_2285_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_2286_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_2287_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(times_times_int(B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_2288_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvd_def
tff(fact_2289_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_2290_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K3: rat] : A = aa_rat_rat(times_times_rat(B),K3) ) ).

% dvd_def
tff(fact_2291_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_2292_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(times_times_int(B),K3) ) ).

% dvd_def
tff(fact_2293_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_2294_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult
tff(fact_2295_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult
tff(fact_2296_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_2297_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),C)) ) ).

% dvd_mult
tff(fact_2298_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_2299_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult2
tff(fact_2300_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult2
tff(fact_2301_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_2302_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),C)) ) ).

% dvd_mult2
tff(fact_2303_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_2304_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_2305_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_2306_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_2307_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_2308_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_2309_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% dvd_triv_left
tff(fact_2310_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B)) ).

% dvd_triv_left
tff(fact_2311_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_2312_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(A),B)) ).

% dvd_triv_left
tff(fact_2313_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_2314_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_2315_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_2316_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_2317_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_2318_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_2319_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_2320_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_2321_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_2322_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_2323_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_2324_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)) ).

% dvd_triv_right
tff(fact_2325_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A)) ).

% dvd_triv_right
tff(fact_2326_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_2327_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),A)) ).

% dvd_triv_right
tff(fact_2328_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)) ) ) ).

% dvd_add
tff(fact_2329_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).

% dvd_add
tff(fact_2330_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C)) ) ) ).

% dvd_add
tff(fact_2331_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_2332_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C)) ) ) ).

% dvd_add
tff(fact_2333_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_2334_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_2335_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_2336_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_2337_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_2338_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_2339_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_2340_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_2341_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_2342_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_2343_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2344_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),A)
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),B)
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2345_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2346_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2347_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2348_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2349_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2350_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),A)
       => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2351_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2352_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2353_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2354_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2355_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_div_div_same
tff(fact_2356_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_2357_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),D)),aa_int_int(divide_divide_int(B),D)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_div_div_same
tff(fact_2358_dvd__power__same,axiom,
    ! [Xa: code_integer,Y: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Nb)) ) ).

% dvd_power_same
tff(fact_2359_dvd__power__same,axiom,
    ! [Xa: nat,Y: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) ) ).

% dvd_power_same
tff(fact_2360_dvd__power__same,axiom,
    ! [Xa: real,Y: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) ) ).

% dvd_power_same
tff(fact_2361_dvd__power__same,axiom,
    ! [Xa: int,Y: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xa),Nb)),aa_nat_int(power_power_int(Y),Nb)) ) ).

% dvd_power_same
tff(fact_2362_dvd__power__same,axiom,
    ! [Xa: complex,Y: complex,Nb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xa),Y)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xa),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) ) ).

% dvd_power_same
tff(fact_2363_gcd__nat_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_2364_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_2365_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_2366_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat)
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_2367_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_2368_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_2369_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_2370_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_2371_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_2372_zero__le,axiom,
    ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa) ).

% zero_le
tff(fact_2373_verit__comp__simplify1_I3_J,axiom,
    ! [B3: real,A3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),A3)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_2374_verit__comp__simplify1_I3_J,axiom,
    ! [B3: extended_enat,A3: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B3),A3)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_2375_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B3),A3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_2376_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A3: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B3),A3)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_2377_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B3),A3)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_2378_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),A3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_2379_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_2380_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),C) = aa_int_int(modulo_modulo_int(A),C) ) ) ).

% mod_mod_cancel
tff(fact_2381_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_2382_dvd__mod,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) ) ) ).

% dvd_mod
tff(fact_2383_dvd__mod,axiom,
    ! [K: int,Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Ma)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(modulo_modulo_int(Ma),Nb)) ) ) ).

% dvd_mod
tff(fact_2384_dvd__mod,axiom,
    ! [K: code_integer,Ma: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Ma)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(Ma,Nb)) ) ) ).

% dvd_mod
tff(fact_2385_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2386_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2387_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2388_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2389_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2390_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2391_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2392_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ia),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Ia),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2393_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2394_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(Ia),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2395_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( ( Ia = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2396_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( ( Ia = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Ia),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2397_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( ( Ia = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2398_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( ( Ia = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(Ia),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2399_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2400_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ia),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Ia),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2401_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2402_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(Ia),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2403_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_2404_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_2405_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_2406_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) ) ) ).

% add_mono
tff(fact_2407_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_left_mono
tff(fact_2408_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_left_mono
tff(fact_2409_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_2410_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)) ) ).

% add_left_mono
tff(fact_2411_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_2412_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_right_mono
tff(fact_2413_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_right_mono
tff(fact_2414_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_2415_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)) ) ).

% add_right_mono
tff(fact_2416_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% le_iff_add
tff(fact_2417_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_2418_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2419_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2420_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_2421_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_2422_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2423_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2424_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_2425_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_2426_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_2427_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_2428_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_2429_le__num__One__iff,axiom,
    ! [Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),one)
    <=> ( Xa = one ) ) ).

% le_num_One_iff
tff(fact_2430_transitive__stepwise__le,axiom,
    ! [Ma: nat,Nb: nat,R3: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(R3,X3),X3)
       => ( ! [X3: nat,Y3: nat,Z4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R3,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R3,Y3),Z4)
               => aa_nat_o(aa_nat_fun_nat_o(R3,X3),Z4) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R3,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R3,Ma),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_2431_nat__induct__at__least,axiom,
    ! [Ma: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_o(P,Ma)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_2432_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
             => aa_nat_o(P,M) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_2433_not__less__eq__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Ma) ) ).

% not_less_eq_eq
tff(fact_2434_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_2435_le__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
        | ( Ma = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_2436_Suc__le__D,axiom,
    ! [Nb: nat,M5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M5)
     => ? [M2: nat] : M5 = aa_nat_nat(suc,M2) ) ).

% Suc_le_D
tff(fact_2437_le__SucI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_2438_le__SucE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( Ma = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_2439_Suc__leD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% Suc_leD
tff(fact_2440_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_2441_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_2442_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_2443_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_2444_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,Ia: nat,J: nat] :
      ( ! [I2: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J3)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Ia)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_2445_le__neq__implies__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( ( Ma != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% le_neq_implies_less
tff(fact_2446_less__or__eq__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Ma = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% less_or_eq_imp_le
tff(fact_2447_le__eq__less__or__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Ma = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_2448_less__imp__le__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% less_imp_le_nat
tff(fact_2449_nat__less__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
        & ( Ma != Nb ) ) ) ).

% nat_less_le
tff(fact_2450_less__eq__real__def,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_real_def
tff(fact_2451_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_2452_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),B2)
    <=> ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),set_complex2(Xs))
         => aa_set_complex_o(member_complex(X4),B2) ) ) ).

% subset_code(1)
tff(fact_2453_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B2)
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),set_real2(Xs))
         => aa_set_real_o(member_real(X4),B2) ) ) ).

% subset_code(1)
tff(fact_2454_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B2)
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),set_set_nat2(Xs))
         => aa_set_set_nat_o(member_set_nat(X4),B2) ) ) ).

% subset_code(1)
tff(fact_2455_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X4,B2) ) ) ).

% subset_code(1)
tff(fact_2456_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B2)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_set_int_o(member_int(X4),B2) ) ) ).

% subset_code(1)
tff(fact_2457_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B2)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X4),B2) ) ) ).

% subset_code(1)
tff(fact_2458_add__leE,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_2459_le__add1,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) ).

% le_add1
tff(fact_2460_le__add2,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% le_add2
tff(fact_2461_add__leD1,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% add_leD1
tff(fact_2462_add__leD2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_2463_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_2464_add__le__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_2465_add__le__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_2466_trans__le__add1,axiom,
    ! [Ia: nat,J: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ma)) ) ).

% trans_le_add1
tff(fact_2467_trans__le__add2,axiom,
    ! [Ia: nat,J: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),J)) ) ).

% trans_le_add2
tff(fact_2468_nat__le__iff__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K3) ) ).

% nat_le_iff_add
tff(fact_2469_mult__le__mono2,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ia)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_2470_mult__le__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_2471_mult__le__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_2472_le__square,axiom,
    ! [Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Ma)) ).

% le_square
tff(fact_2473_le__cube,axiom,
    ! [Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Ma))) ).

% le_cube
tff(fact_2474_div__le__mono,axiom,
    ! [Ma: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).

% div_le_mono
tff(fact_2475_div__le__dividend,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Ma) ).

% div_le_dividend
tff(fact_2476_subset__iff__psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_set_nat(A2),B2)
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_2477_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_set_nat(B2),C5)
       => aa_set_nat_o(ord_less_set_nat(A2),C5) ) ) ).

% subset_psubset_trans
tff(fact_2478_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% subset_not_subset_eq
tff(fact_2479_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C5)
       => aa_set_nat_o(ord_less_set_nat(A2),C5) ) ) ).

% psubset_subset_trans
tff(fact_2480_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% psubset_imp_subset
tff(fact_2481_psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_2482_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
         => aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% psubsetE
tff(fact_2483_mod__less__eq__dividend,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)),Ma) ).

% mod_less_eq_dividend
tff(fact_2484_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Nb),zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_2485_i0__lb,axiom,
    ! [Nb: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),Nb) ).

% i0_lb
tff(fact_2486_dvd__power__iff,axiom,
    ! [Xa: code_integer,Ma: nat,Nb: nat] :
      ( ( Xa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Ma)),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_2487_dvd__power__iff,axiom,
    ! [Xa: nat,Ma: nat,Nb: nat] :
      ( ( Xa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_2488_dvd__power__iff,axiom,
    ! [Xa: int,Ma: nat,Nb: nat] :
      ( ( Xa != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xa),Ma)),aa_nat_int(power_power_int(Xa),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_2489_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2490_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2491_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2492_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2493_power__le__imp__le__exp,axiom,
    ! [A: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2494_power__le__imp__le__exp,axiom,
    ! [A: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Ma)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2495_power__le__imp__le__exp,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2496_power__le__imp__le__exp,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Ma)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2497_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_2498_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F,some_num(A),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_2499_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_2500_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_2501_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se4203085406695923979it_int(Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_2502_set__bit__greater__eq,axiom,
    ! [K: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se7879613467334960850it_int(Nb),K)) ).

% set_bit_greater_eq
tff(fact_2503_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_2504_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] : aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_2505_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_2506_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_2507_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_2508_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_2509_dvd__power__iff__le,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_2510_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,aa_o_complex(zero_n1201886186963655149omplex,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P2)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_2511_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,aa_o_real(zero_n3304061248610475627l_real,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P2)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_2512_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,aa_o_rat(zero_n2052037380579107095ol_rat,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P2)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_2513_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P2)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_2514_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P2)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_2515_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P2)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_2516_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,aa_o_complex(zero_n1201886186963655149omplex,(P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_2517_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,aa_o_real(zero_n3304061248610475627l_real,(P2)))
    <=> ( ( (P2)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P2)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_2518_split__of__bool,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,aa_o_rat(zero_n2052037380579107095ol_rat,(P2)))
    <=> ( ( (P2)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P2)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_2519_split__of__bool,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ( ( (P2)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P2)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_2520_split__of__bool,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ( ( (P2)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P2)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_2521_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ( ( (P2)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P2)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_2522_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_complex(zero_n1201886186963655149omplex,(P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_2523_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_real(zero_n3304061248610475627l_real,(P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_2524_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_rat(zero_n2052037380579107095ol_rat,(P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_2525_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_2526_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P2)) = $ite((P2),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_2527_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n356916108424825756nteger((P2)) = $ite((P2),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_2528_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_2529_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_2530_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_2531_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2532_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2533_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2534_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2535_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2536_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2537_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_2538_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_2539_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(times_times_int(B),A) = aa_int_int(times_times_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_2540_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_2541_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_2542_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(times_times_int(A),B) = aa_int_int(times_times_int(A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_2543_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2544_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2545_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2546_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2547_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2548_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2549_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_2550_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_2551_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_2552_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_2553_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_2554_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_2555_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_2556_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_2557_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_2558_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2559_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2560_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(C),D)) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2561_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_2562_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_2563_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)) ) ).

% dvd_mult_imp_div
tff(fact_2564_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_2565_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_2566_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(B),C)),A)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_2567_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_2568_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_2569_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_2570_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% div_mult_swap
tff(fact_2571_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_2572_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(times_times_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),C) ) ) ).

% div_mult_swap
tff(fact_2573_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_2574_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_2575_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(B),C)),A) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_2576_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2577_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2578_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2579_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2580_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2581_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2582_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_2583_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_2584_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_2585_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_2586_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_2587_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_2588_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_2589_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_2590_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_2591_div__power,axiom,
    ! [B: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_2592_div__power,axiom,
    ! [B: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_2593_div__power,axiom,
    ! [B: int,A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(power_power_int(aa_int_int(divide_divide_int(A),B)),Nb) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% div_power
tff(fact_2594_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_2595_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_2596_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_2597_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2598_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2599_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2600_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_2601_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_2602_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_2603_zero__le__numeral,axiom,
    ! [Nb: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)) ).

% zero_le_numeral
tff(fact_2604_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_le_numeral
tff(fact_2605_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_le_numeral
tff(fact_2606_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_2607_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_2608_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)),zero_z5237406670263579293d_enat) ).

% not_numeral_le_zero
tff(fact_2609_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_2610_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_2611_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_2612_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_2613_dvd__pos__nat,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma) ) ) ).

% dvd_pos_nat
tff(fact_2614_nat__dvd__not__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma) ) ) ).

% nat_dvd_not_less
tff(fact_2615_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2616_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2617_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2618_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2619_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_2620_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_2621_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_2622_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2623_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2624_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2625_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2626_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_2627_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2628_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2629_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_2630_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_2631_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2632_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2633_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_2634_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2635_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2636_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2637_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2638_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_2639_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_2640_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_2641_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_2642_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2643_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2644_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2645_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono
tff(fact_2646_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono
tff(fact_2647_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_2648_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ) ) ).

% mult_right_mono
tff(fact_2649_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2650_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2651_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2652_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono
tff(fact_2653_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono
tff(fact_2654_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_2655_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% mult_left_mono
tff(fact_2656_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2657_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2658_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2659_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2660_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2661_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2662_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_2663_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ).

% split_mult_pos_le
tff(fact_2664_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ).

% split_mult_pos_le
tff(fact_2665_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).

% zero_le_square
tff(fact_2666_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).

% zero_le_square
tff(fact_2667_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),A)) ).

% zero_le_square
tff(fact_2668_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2669_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2670_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2671_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2672_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2673_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2674_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2675_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2676_add__nonpos__eq__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y) = zero_zero_real )
        <=> ( ( Xa = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2677_add__nonpos__eq__0__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xa),Y) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2678_add__nonpos__eq__0__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2679_add__nonpos__eq__0__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(plus_plus_int(Xa),Y) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2680_add__nonneg__eq__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y) = zero_zero_real )
        <=> ( ( Xa = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2681_add__nonneg__eq__0__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(plus_plus_rat(Xa),Y) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2682_add__nonneg__eq__0__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2683_add__nonneg__eq__0__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(plus_plus_int(Xa),Y) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2684_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_2685_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_2686_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_2687_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_2688_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2689_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2690_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2691_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2692_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_2693_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_increasing2
tff(fact_2694_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_2695_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% add_increasing2
tff(fact_2696_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2697_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2698_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2699_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2700_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_2701_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_increasing
tff(fact_2702_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_2703_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% add_increasing
tff(fact_2704_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2705_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing
tff(fact_2706_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2707_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),B) ) ) ).

% add_decreasing
tff(fact_2708_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_2709_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_2710_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_2711_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_2712_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2713_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2714_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2715_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2716_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_2717_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_2718_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_2719_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_2720_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2721_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_2722_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2723_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) ) ) ).

% add_less_le_mono
tff(fact_2724_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2725_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_2726_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2727_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) ) ) ).

% add_le_less_mono
tff(fact_2728_add__mono__thms__linordered__field_I3_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ia),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2729_add__mono__thms__linordered__field_I3_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ia),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Ia),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2730_add__mono__thms__linordered__field_I3_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2731_add__mono__thms__linordered__field_I3_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Ia),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2732_add__mono__thms__linordered__field_I4_J,axiom,
    ! [Ia: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2733_add__mono__thms__linordered__field_I4_J,axiom,
    ! [Ia: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ia),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Ia),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2734_add__mono__thms__linordered__field_I4_J,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2735_add__mono__thms__linordered__field_I4_J,axiom,
    ! [Ia: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Ia),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2736_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2737_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2738_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_right_mono
tff(fact_2739_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_right_mono
tff(fact_2740_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_2741_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_2742_divide__nonneg__nonneg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2743_divide__nonneg__nonneg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2744_divide__nonneg__nonpos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2745_divide__nonneg__nonpos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2746_divide__nonpos__nonneg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2747_divide__nonpos__nonneg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2748_divide__nonpos__nonpos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2749_divide__nonpos__nonpos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2750_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2751_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2752_one__le__numeral,axiom,
    ! [Nb: num] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,one_on7984719198319812577d_enat),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)) ).

% one_le_numeral
tff(fact_2753_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).

% one_le_numeral
tff(fact_2754_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_le_numeral
tff(fact_2755_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_2756_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_2757_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_2758_power__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_2759_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_2760_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono
tff(fact_2761_zero__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_power
tff(fact_2762_zero__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_power
tff(fact_2763_zero__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_le_power
tff(fact_2764_zero__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_power
tff(fact_2765_one__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% one_le_power
tff(fact_2766_one__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% one_le_power
tff(fact_2767_one__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% one_le_power
tff(fact_2768_one__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% one_le_power
tff(fact_2769_zdvd__not__zless,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ma),Nb)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Ma) ) ) ).

% zdvd_not_zless
tff(fact_2770_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D3) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_2771_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xa) = 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),Xa) = 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] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),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_2772_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2773_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2774_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2775_zdvd__mult__cancel,axiom,
    ! [K: int,Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(K),Ma)),aa_int_int(times_times_int(K),Nb))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ma),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_2776_Suc__leI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb) ) ).

% Suc_leI
tff(fact_2777_Suc__le__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_le_eq
tff(fact_2778_dec__induct,axiom,
    ! [Ia: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(P,Ia)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_2779_inc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,Ia) ) ) ) ).

% inc_induct
tff(fact_2780_Suc__le__lessD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_le_lessD
tff(fact_2781_le__less__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma))
      <=> ( Nb = Ma ) ) ) ).

% le_less_Suc_eq
tff(fact_2782_less__Suc__eq__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% less_Suc_eq_le
tff(fact_2783_less__eq__Suc__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Ma) ) ).

% less_eq_Suc_le
tff(fact_2784_le__imp__less__Suc,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_2785_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_2786_dbl__def,axiom,
    ! [Xa: real] : aa_real_real(neg_numeral_dbl_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Xa) ).

% dbl_def
tff(fact_2787_dbl__def,axiom,
    ! [Xa: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xa) = aa_rat_rat(plus_plus_rat(Xa),Xa) ).

% dbl_def
tff(fact_2788_dbl__def,axiom,
    ! [Xa: int] : aa_int_int(neg_numeral_dbl_int,Xa) = aa_int_int(plus_plus_int(Xa),Xa) ).

% dbl_def
tff(fact_2789_zdvd__period,axiom,
    ! [A: int,D: int,Xa: int,Ta: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(Xa),Ta))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(Xa),aa_int_int(times_times_int(C),D))),Ta)) ) ) ).

% zdvd_period
tff(fact_2790_zdvd__reduce,axiom,
    ! [K: int,Nb: int,Ma: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(plus_plus_int(Nb),aa_int_int(times_times_int(K),Ma)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb) ) ).

% zdvd_reduce
tff(fact_2791_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Ma: nat,K: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M2)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Ma)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K))) ) ).

% mono_nat_linear_lb
tff(fact_2792_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_2793_Suc__div__le__mono,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Ma)),Nb)) ).

% Suc_div_le_mono
tff(fact_2794_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_2795_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb))),Ma) ).

% times_div_less_eq_dividend
tff(fact_2796_div__times__less__eq__dividend,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Nb)),Ma) ).

% div_times_less_eq_dividend
tff(fact_2797_mod__Suc__le__divisor,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_2798_zmod__le__nonneg__dividend,axiom,
    ! [Ma: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ma)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(Ma),K)),Ma) ) ).

% zmod_le_nonneg_dividend
tff(fact_2799_zero__le__even__power,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_even_power
tff(fact_2800_zero__le__even__power,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_even_power
tff(fact_2801_zero__le__even__power,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_even_power
tff(fact_2802_zero__le__odd__power,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_2803_zero__le__odd__power,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_2804_zero__le__odd__power,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_2805_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_2806_zero__le__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_2807_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_2808_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_2809_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_2810_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_2811_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xa: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xa,Xaa,Xb) = Y )
     => ( ( ( Xaa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V2)
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A4: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A4) )
               => ! [B4: product_prod_nat_nat] :
                    ( ( Xb = some_P7363390416028606310at_nat(B4) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xa,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_2812_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xa: fun_num_fun_num_num,Xaa: option_num,Xb: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(Xa,Xaa,Xb) = Y )
     => ( ( ( Xaa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xaa = some_num(V2)
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A4: num] :
                ( ( Xaa = some_num(A4) )
               => ! [B4: num] :
                    ( ( Xb = some_num(B4) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(Xa,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_2813_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xa: fun_nat_fun_nat_nat,Xaa: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xa),Xaa),Xb) = Y )
     => ( ( ( Xaa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xaa = some_nat(V2)
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A4: nat] :
                ( ( Xaa = some_nat(A4) )
               => ! [B4: nat] :
                    ( ( Xb = some_nat(B4) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xa,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_2814_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).

% unit_dvdE
tff(fact_2815_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_2816_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] : B != aa_int_int(times_times_int(A),C2) ) ) ).

% unit_dvdE
tff(fact_2817_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2818_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2819_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(D),C) )
            <=> ( aa_int_int(times_times_int(B),C) = aa_int_int(times_times_int(A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2820_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2821_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2822_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2823_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2824_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2825_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2826_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2827_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2828_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(divide_divide_int(B),A) = C )
        <=> ( B = aa_int_int(times_times_int(C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2829_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2830_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2831_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2832_even__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_2833_even__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_2834_even__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_2835_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2836_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2837_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2838_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_2839_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_2840_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(times_times_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_2841_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).

% unit_div_commute
tff(fact_2842_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_2843_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),C) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),C)),B) ) ) ).

% unit_div_commute
tff(fact_2844_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_2845_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_2846_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_2847_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_2848_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_2849_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(divide_divide_int(C),B) )
      <=> ( aa_int_int(times_times_int(A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_2850_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_2851_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_2852_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = C )
      <=> ( A = aa_int_int(times_times_int(C),B) ) ) ) ).

% unit_eq_div1
tff(fact_2853_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_2854_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_2855_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_2856_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_2857_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_2858_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_2859_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2860_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2861_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2862_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2863_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2864_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2865_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2866_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2867_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2868_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2869_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2870_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2871_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2872_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2873_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2874_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2875_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2876_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2877_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2878_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2879_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2880_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2881_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2882_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2883_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2884_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2885_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2886_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2887_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2888_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2889_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2890_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2891_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2892_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2893_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2894_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2895_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2896_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2897_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2898_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2899_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2900_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2901_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2902_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2903_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2904_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2905_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2906_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2907_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2908_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2909_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2910_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xa),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2911_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Xa),Z)),aa_int_int(times_times_int(Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2912_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2913_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Xa)),aa_rat_rat(times_times_rat(Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2914_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Z),Xa)),aa_int_int(times_times_int(Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2915_field__le__epsilon,axiom,
    ! [Xa: real,Y: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% field_le_epsilon
tff(fact_2916_field__le__epsilon,axiom,
    ! [Xa: rat,Y: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(plus_plus_rat(Y),E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% field_le_epsilon
tff(fact_2917_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2918_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2919_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2920_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2921_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2922_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing
tff(fact_2923_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2924_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% add_strict_increasing
tff(fact_2925_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2926_add__pos__nonneg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),plus_p3455044024723400733d_enat(A,B)) ) ) ).

% add_pos_nonneg
tff(fact_2927_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2928_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2929_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2930_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_2931_add__nonpos__neg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),zero_z5237406670263579293d_enat)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,plus_p3455044024723400733d_enat(A,B)),zero_z5237406670263579293d_enat) ) ) ).

% add_nonpos_neg
tff(fact_2932_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2933_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2934_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2935_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2936_add__nonneg__pos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),plus_p3455044024723400733d_enat(A,B)) ) ) ).

% add_nonneg_pos
tff(fact_2937_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2938_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2939_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2940_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_2941_add__neg__nonpos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),zero_z5237406670263579293d_enat)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),zero_z5237406670263579293d_enat)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,plus_p3455044024723400733d_enat(A,B)),zero_z5237406670263579293d_enat) ) ) ).

% add_neg_nonpos
tff(fact_2942_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2943_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2944_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2945_frac__le,axiom,
    ! [Y: real,Xa: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_le
tff(fact_2946_frac__le,axiom,
    ! [Y: rat,Xa: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xa),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_le
tff(fact_2947_frac__less,axiom,
    ! [Xa: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less
tff(fact_2948_frac__less,axiom,
    ! [Xa: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xa),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_less
tff(fact_2949_frac__less2,axiom,
    ! [Xa: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_2950_frac__less2,axiom,
    ! [Xa: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xa),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_2951_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2952_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2953_divide__nonneg__neg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_2954_divide__nonneg__neg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_2955_divide__nonneg__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2956_divide__nonneg__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2957_divide__nonpos__neg,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2958_divide__nonpos__neg,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2959_divide__nonpos__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_2960_divide__nonpos__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_2961_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2962_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2963_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_2964_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_positive
tff(fact_2965_sum__squares__le__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2966_sum__squares__le__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xa),Xa)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2967_sum__squares__le__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xa),Xa)),aa_int_int(times_times_int(Y),Y))),zero_zero_int)
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2968_sum__squares__ge__zero,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2969_sum__squares__ge__zero,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xa),Xa)),aa_rat_rat(times_times_rat(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2970_sum__squares__ge__zero,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xa),Xa)),aa_int_int(times_times_int(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2971_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2972_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2973_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2974_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2975_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2976_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2977_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_2978_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_2979_mult__right__le__one__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2980_mult__right__le__one__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2981_mult__right__le__one__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_2982_mult__left__le__one__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2983_mult__left__le__one__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2984_mult__left__le__one__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_2985_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2986_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2987_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2988_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2989_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2990_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2991_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_2992_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),one_one_int)),B) ) ).

% discrete
tff(fact_2993_power__le__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_2994_power__le__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_2995_power__le__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2996_power__le__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_2997_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2998_dvd__mult__cancel,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_2999_dvd__mult__cancel2,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Ma)),Ma)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3000_dvd__mult__cancel1,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),Ma)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3001_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3002_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3003_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3004_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3005_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3006_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3007_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3008_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3009_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3010_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3011_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3012_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3013_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3014_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3015_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X3: nat,Y3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3016_mod__greater__zero__iff__not__dvd,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3017_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),K2)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_3018_nat__mult__le__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_3019_div__le__mono2,axiom,
    ! [Ma: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Ma)) ) ) ).

% div_le_mono2
tff(fact_3020_div__greater__zero__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_3021_nat__one__le__power,axiom,
    ! [Ia: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Ia)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ia),Nb)) ) ).

% nat_one_le_power
tff(fact_3022_mod__le__divisor,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_3023_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_3024_zdiv__mono1,axiom,
    ! [A: int,A3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A3),B)) ) ) ).

% zdiv_mono1
tff(fact_3025_zdiv__mono2,axiom,
    ! [A: int,B3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B3)) ) ) ) ).

% zdiv_mono2
tff(fact_3026_zdiv__eq__0__iff,axiom,
    ! [Ia: int,K: int] :
      ( ( aa_int_int(divide_divide_int(Ia),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ia)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),Ia) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3027_zdiv__mono1__neg,axiom,
    ! [A: int,A3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A3),B)),aa_int_int(divide_divide_int(A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_3028_zdiv__mono2__neg,axiom,
    ! [A: int,B3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B3)),aa_int_int(divide_divide_int(A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3029_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_3030_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(K),L)) ) ) ).

% div_positive_int
tff(fact_3031_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3032_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3033_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,Ia: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(Ia),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),Ia) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3034_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3035_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3036_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3037_mod__eq__nat1E,axiom,
    ! [Ma: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ~ ! [S: nat] : Ma != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat1E
tff(fact_3038_mod__eq__nat2E,axiom,
    ! [Ma: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ~ ! [S: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat2E
tff(fact_3039_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
       => ? [Q3: nat] : Xa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3040_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3041_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3042_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_3043_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% neg_mod_conj
tff(fact_3044_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_3045_zmod__trivial__iff,axiom,
    ! [Ia: int,K: int] :
      ( ( aa_int_int(modulo_modulo_int(Ia),K) = Ia )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ia)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),Ia) ) ) ) ).

% zmod_trivial_iff
tff(fact_3046_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3047_power__le__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3048_power__le__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3049_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_3050_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_3051_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_3052_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_3053_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_3054_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_3055_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3056_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3057_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3058_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3059_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3060_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3061_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B4: code_integer] :
              ( ( B4 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B4),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B4 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B4) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B4) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3062_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B4),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B4 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B4) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B4) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3063_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B4),one_one_int)
               => ( ( aa_int_int(divide_divide_int(one_one_int),A) = B4 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B4) = A )
                   => ( ( aa_int_int(times_times_int(A),B4) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(times_times_int(C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3064_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_3065_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_3066_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B4: int] : A != aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_3067_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_3068_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_3069_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A),B)) ) ) ).

% odd_even_add
tff(fact_3070_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_3071_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_3072_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_3073_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) )
        & ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_3074_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_3075_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) )
        & ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_3076_dvd__power,axiom,
    ! [Nb: nat,Xa: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xa = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xa),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),Nb)) ) ).

% dvd_power
tff(fact_3077_dvd__power,axiom,
    ! [Nb: nat,Xa: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xa = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xa),aa_nat_rat(power_power_rat(Xa),Nb)) ) ).

% dvd_power
tff(fact_3078_dvd__power,axiom,
    ! [Nb: nat,Xa: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xa = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Nb)) ) ).

% dvd_power
tff(fact_3079_dvd__power,axiom,
    ! [Nb: nat,Xa: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xa = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)) ) ).

% dvd_power
tff(fact_3080_dvd__power,axiom,
    ! [Nb: nat,Xa: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xa = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xa),aa_nat_int(power_power_int(Xa),Nb)) ) ).

% dvd_power
tff(fact_3081_dvd__power,axiom,
    ! [Nb: nat,Xa: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xa = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xa),aa_nat_complex(power_power_complex(Xa),Nb)) ) ).

% dvd_power
tff(fact_3082_field__le__mult__one__interval,axiom,
    ! [Xa: real,Y: real] :
      ( ! [Z4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z4),Xa)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% field_le_mult_one_interval
tff(fact_3083_field__le__mult__one__interval,axiom,
    ! [Xa: rat,Y: rat] :
      ( ! [Z4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z4),Xa)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% field_le_mult_one_interval
tff(fact_3084_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3085_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3086_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3087_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3088_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3089_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3090_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3091_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3092_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3093_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3094_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3095_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3096_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3097_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3098_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3099_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3100_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3101_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3102_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3103_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3104_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3105_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3106_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3107_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3108_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_3109_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3110_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_3111_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3112_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3113_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3114_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3115_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3116_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3117_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3118_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3119_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3120_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3121_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3122_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xa: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3123_mult__imp__div__pos__le,axiom,
    ! [Y: rat,Xa: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3124_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3125_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),Xa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3126_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3127_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3128_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3129_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3130_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3131_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3132_even__signed__take__bit__iff,axiom,
    ! [Ma: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Ma,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_3133_even__signed__take__bit__iff,axiom,
    ! [Ma: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Ma),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_3134_convex__bound__le,axiom,
    ! [Xa: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3135_convex__bound__le,axiom,
    ! [Xa: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xa)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3136_convex__bound__le,axiom,
    ! [Xa: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(plus_plus_int(U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),Xa)),aa_int_int(times_times_int(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3137_power__Suc__le__self,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3138_power__Suc__le__self,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3139_power__Suc__le__self,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3140_power__Suc__le__self,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3141_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3142_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3143_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3144_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3145_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3146_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3147_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3148_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3149_self__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% self_le_power
tff(fact_3150_self__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% self_le_power
tff(fact_3151_self__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% self_le_power
tff(fact_3152_self__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% self_le_power
tff(fact_3153_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_3154_power2__nat__le__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_3155_power2__nat__le__eq__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_3156_self__le__ge2__pow,axiom,
    ! [K: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Ma)) ) ).

% self_le_ge2_pow
tff(fact_3157_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_3158_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_3159_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A),(B))) = $ite(
        (A),
        some_nat(zero_zero_nat),
        $ite((B),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_3160_vebt__maxt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A),(B))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_3161_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_3162_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_3163_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3164_q__pos__lemma,axiom,
    ! [B3: int,Q5: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_3165_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q5: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3166_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q5: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B3),Q5)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3167_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q5)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3168_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q5)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3169_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),L)),zero_zero_int)
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(plus_plus_int(K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3170_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_3171_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_3172_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_3173_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3174_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3175_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3176_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3177_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3178_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3179_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_3180_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3181_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3182_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3183_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3184_convex__bound__lt,axiom,
    ! [Xa: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3185_convex__bound__lt,axiom,
    ! [Xa: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xa)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3186_convex__bound__lt,axiom,
    ! [Xa: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(plus_plus_int(U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),Xa)),aa_int_int(times_times_int(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3187_power2__le__imp__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3188_power2__le__imp__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3189_power2__le__imp__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3190_power2__le__imp__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ).

% power2_le_imp_le
tff(fact_3191_power2__eq__imp__eq,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3192_power2__eq__imp__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3193_power2__eq__imp__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3194_power2__eq__imp__eq,axiom,
    ! [Xa: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xa = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3195_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3196_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3197_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3198_even__unset__bit__iff,axiom,
    ! [Ma: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Ma,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( Ma = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3199_even__unset__bit__iff,axiom,
    ! [Ma: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(Ma),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( Ma = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3200_even__unset__bit__iff,axiom,
    ! [Ma: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( Ma = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3201_even__set__bit__iff,axiom,
    ! [Ma: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Ma,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( Ma != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3202_even__set__bit__iff,axiom,
    ! [Ma: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(Ma),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( Ma != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3203_even__set__bit__iff,axiom,
    ! [Ma: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Ma),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( Ma != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3204_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3205_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3206_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3207_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3208_even__flip__bit__iff,axiom,
    ! [Ma: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Ma,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( Ma = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3209_even__flip__bit__iff,axiom,
    ! [Ma: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Ma),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( Ma = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3210_even__flip__bit__iff,axiom,
    ! [Ma: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Ma),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( Ma = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3211_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3212_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3213_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3214_split__div_H,axiom,
    ! [P: fun_nat_o,Ma: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_3215_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_3216_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(divide_divide_int(Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I)),J2) ) )
             => aa_int_o(P,I) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I)),J2) ) )
             => aa_int_o(P,I) ) ) ) ) ).

% split_zdiv
tff(fact_3217_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(divide_divide_int(A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3218_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(divide_divide_int(A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3219_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(modulo_modulo_int(Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I)),J2) ) )
             => aa_int_o(P,J2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I)),J2) ) )
             => aa_int_o(P,J2) ) ) ) ) ).

% split_zmod
tff(fact_3220_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3221_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3222_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A2),Nb)),
                $ite(aa_int_int(modulo_modulo_int(B2),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(divide_divide_int(B2),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_3223_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3224_bits__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [A4: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
         => aa_nat_o(P,A4) )
     => ( ! [A4: nat,B4: $o] :
            ( aa_nat_o(P,A4)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B4))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B4))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_3225_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A4: int] :
          ( ( aa_int_int(divide_divide_int(A4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
         => aa_int_o(P,A4) )
     => ( ! [A4: int,B4: $o] :
            ( aa_int_o(P,A4)
           => ( ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(B4))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
             => aa_int_o(P,aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(B4))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_3226_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( divide6298287555418463151nteger(A4,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A4 )
         => aa_Code_integer_o(P,A4) )
     => ( ! [A4: code_integer,B4: $o] :
            ( aa_Code_integer_o(P,A4)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A4)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A4 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A4))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_3227_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B4: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B4),one_one_Code_integer) ) ).

% oddE
tff(fact_3228_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B4: 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))),B4)),one_one_nat) ) ).

% oddE
tff(fact_3229_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B4: int] : A != aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4)),one_one_int) ) ).

% oddE
tff(fact_3230_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_3231_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3232_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_3233_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_3234_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_3235_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_3236_power2__less__imp__less,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3237_power2__less__imp__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3238_power2__less__imp__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3239_power2__less__imp__less,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ) ).

% power2_less_imp_less
tff(fact_3240_sum__power2__le__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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)
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3241_sum__power2__le__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xa),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)
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3242_sum__power2__le__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xa),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)
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3243_sum__power2__ge__zero,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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_3244_sum__power2__ge__zero,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xa),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_3245_sum__power2__ge__zero,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xa),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_3246_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3247_zero__le__even__power_H,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3248_zero__le__even__power_H,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3249_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_3250_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_3251_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_3252_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Nb),K)),aa_int_int(modulo_modulo_int(Nb),K))
      <=> ! [I: int,J2: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
              & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I)),J2) ) )
           => aa_int_o(aa_int_fun_int_o(P,I),J2) ) ) ) ).

% split_neg_lemma
tff(fact_3253_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Nb),K)),aa_int_int(modulo_modulo_int(Nb),K))
      <=> ! [I: int,J2: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
              & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I)),J2) ) )
           => aa_int_o(aa_int_fun_int_o(P,I),J2) ) ) ) ).

% split_pos_lemma
tff(fact_3254_exp__mod__exp,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ).

% exp_mod_exp
tff(fact_3255_exp__mod__exp,axiom,
    ! [Ma: nat,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) ).

% exp_mod_exp
tff(fact_3256_exp__mod__exp,axiom,
    ! [Ma: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) ).

% exp_mod_exp
tff(fact_3257_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3258_zero__less__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3259_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3260_sum__squares__bound,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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_3261_sum__squares__bound,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xa)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xa),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_3262_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3263_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3264_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3265_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3266_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3267_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3268_mod__double__modulus,axiom,
    ! [Ma: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ma)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
       => ( ( modulo364778990260209775nteger(Xa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) = modulo364778990260209775nteger(Xa,Ma) )
          | ( modulo364778990260209775nteger(Xa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xa,Ma),Ma) ) ) ) ) ).

% mod_double_modulus
tff(fact_3269_mod__double__modulus,axiom,
    ! [Ma: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ma) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Ma)),Ma) ) ) ) ) ).

% mod_double_modulus
tff(fact_3270_mod__double__modulus,axiom,
    ! [Ma: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( ( aa_int_int(modulo_modulo_int(Xa),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) = aa_int_int(modulo_modulo_int(Xa),Ma) )
          | ( aa_int_int(modulo_modulo_int(Xa),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(Xa),Ma)),Ma) ) ) ) ) ).

% mod_double_modulus
tff(fact_3271_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] :
      vEBT_VEBT_insert(vEBT_Node(Info,Dega,TreeLista,Summarya),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)),Xa),vEBT_Node(Info,Dega,TreeLista,Summarya),vEBT_vebt_insert(vEBT_Node(Info,Dega,TreeLista,Summarya),Xa)) ).

% VEBT_internal.insert'.simps(2)
tff(fact_3272_less__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(Xa)),some_nat(Y)) ) ).

% less_shift
tff(fact_3273_greater__shift,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(Xa)),some_nat(Y)) ) ).

% greater_shift
tff(fact_3274_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xa) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpypredd
tff(fact_3275_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xa) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpyd
tff(fact_3276_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_3277_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(plus_plus_int(X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X: int] :
              ( aa_int_o(P,X)
             => aa_int_o(P,aa_int_int(plus_plus_int(X),aa_int_int(times_times_int(K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3278_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_3279_div2__even__ext__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( Xa = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_3280_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X4: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X4))
    <=> ? [X4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X4,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3281_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X4: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X4))
    <=> ? [X4: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X4),zero_zero_complex))
          & aa_complex_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3282_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X4: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X4))
    <=> ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),zero_zero_real))
          & aa_real_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3283_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X4),zero_zero_rat))
          & aa_rat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3284_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X4))
    <=> ? [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),zero_zero_nat))
          & aa_nat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3285_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(P,aa_int_int(times_times_int(L),X4))
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(plus_plus_int(X4),zero_zero_int))
          & aa_int_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3286_subset__antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
       => ( A2 = B2 ) ) ) ).

% subset_antisym
tff(fact_3287_subsetI,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => aa_set_complex_o(member_complex(X3),B2) )
     => aa_set_complex_o(ord_le211207098394363844omplex(A2),B2) ) ).

% subsetI
tff(fact_3288_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_set_real_o(member_real(X3),B2) )
     => aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ).

% subsetI
tff(fact_3289_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A2)
         => aa_set_set_nat_o(member_set_nat(X3),B2) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ).

% subsetI
tff(fact_3290_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_set_int_o(member_int(X3),B2) )
     => aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% subsetI
tff(fact_3291_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_set_nat_o(member_nat(X3),B2) )
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% subsetI
tff(fact_3292_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xa),Sx) ) ) ).

% succ_corr
tff(fact_3293_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Px: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xa) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xa),Px) ) ) ).

% pred_corr
tff(fact_3294_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(Ta)),Xa),Sx) ) ) ).

% succ_correct
tff(fact_3295_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(Ta)),Xa),Sx) ) ) ).

% pred_correct
tff(fact_3296_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_3297_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_3298_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_3299_diff__self,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),A) = zero_zero_int ).

% diff_self
tff(fact_3300_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_3301_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_3302_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_3303_diff__0__right,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),zero_zero_int) = A ).

% diff_0_right
tff(fact_3304_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_3305_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_3306_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_3307_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_3308_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_3309_diff__zero,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),zero_zero_int) = A ).

% diff_zero
tff(fact_3310_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_3311_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3312_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3313_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_3314_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3315_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3316_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3317_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_3318_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3319_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_3320_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_3321_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3322_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(A),B) ).

% add_diff_cancel_right
tff(fact_3323_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3324_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3325_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_3326_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3327_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_3328_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_3329_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3330_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)) = aa_int_int(minus_minus_int(A),B) ).

% add_diff_cancel_left
tff(fact_3331_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3332_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3333_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3334_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3335_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3336_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3337_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mod_self2
tff(fact_3338_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_3339_diff__Suc__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ).

% diff_Suc_Suc
tff(fact_3340_Suc__diff__diff,axiom,
    ! [Ma: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),K) ).

% Suc_diff_diff
tff(fact_3341_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_3342_diff__self__eq__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Ma) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_3343_diff__diff__cancel,axiom,
    ! [Ia: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ia)) = Ia ) ) ).

% diff_diff_cancel
tff(fact_3344_diff__diff__left,axiom,
    ! [Ia: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_3345_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3346_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3347_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3348_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3349_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3350_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3351_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3352_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3353_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3354_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3355_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3356_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3357_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3358_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(plus_plus_int(B),aa_int_int(minus_minus_int(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3359_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_3360_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_3361_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_3362_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_3363_diff__numeral__special_I9_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_3364_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3365_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3366_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3367_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3368_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_3369_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_3370_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_3371_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_3372_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_diff
tff(fact_3373_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_diff
tff(fact_3374_zero__less__diff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% zero_less_diff
tff(fact_3375_diff__is__0__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% diff_is_0_eq
tff(fact_3376_diff__is__0__eq_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_3377_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_3378_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),Ia) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ia)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_3379_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_3380_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_3381_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_3382_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),Ia) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Ia)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_3383_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_3384_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_3385_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_3386_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(minus_minus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A),B)) ) ).

% even_diff
tff(fact_3387_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_3388_even__diff__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ) ) ).

% even_diff_nat
tff(fact_3389_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3390_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3391_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3392_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_3393_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X: real] : aa_set_real_o(member_real(X),S2)
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( aa_set_real_o(member_real(X3),S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z5) )
       => ? [Y3: real] :
            ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y3) )
            & ! [Z5: real] :
                ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),S2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z5) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z5) ) ) ) ) ).

% complete_real
tff(fact_3394_verit__la__generic,axiom,
    ! [A: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
      | ( A = Xa )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A) ) ).

% verit_la_generic
tff(fact_3395_dvd__diff__nat,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_3396_dvd__antisym,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma)
       => ( Ma = Nb ) ) ) ).

% dvd_antisym
tff(fact_3397_Collect__mono__iff,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(collect_real(P)),collect_real(Q))
    <=> ! [X4: real] :
          ( aa_real_o(P,X4)
         => aa_real_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_3398_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q))
    <=> ! [X4: list_nat] :
          ( aa_list_nat_o(P,X4)
         => aa_list_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_3399_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q))
    <=> ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
         => aa_set_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_3400_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q))
    <=> ! [X4: int] :
          ( aa_int_o(P,X4)
         => aa_int_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_3401_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q))
    <=> ! [X4: nat] :
          ( aa_nat_o(P,X4)
         => aa_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_3402_set__eq__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% set_eq_subset
tff(fact_3403_subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C5)
       => aa_set_nat_o(ord_less_eq_set_nat(A2),C5) ) ) ).

% subset_trans
tff(fact_3404_Collect__mono,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X3: real] :
          ( aa_real_o(P,X3)
         => aa_real_o(Q,X3) )
     => aa_set_real_o(ord_less_eq_set_real(collect_real(P)),collect_real(Q)) ) ).

% Collect_mono
tff(fact_3405_Collect__mono,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(P,X3)
         => aa_list_nat_o(Q,X3) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_3406_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
         => aa_set_nat_o(Q,X3) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P)),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_3407_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
         => aa_int_o(Q,X3) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_3408_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
         => aa_nat_o(Q,X3) )
     => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_3409_subset__refl,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A2),A2) ).

% subset_refl
tff(fact_3410_subset__iff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
    <=> ! [T: complex] :
          ( aa_set_complex_o(member_complex(T),A2)
         => aa_set_complex_o(member_complex(T),B2) ) ) ).

% subset_iff
tff(fact_3411_subset__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ! [T: real] :
          ( aa_set_real_o(member_real(T),A2)
         => aa_set_real_o(member_real(T),B2) ) ) ).

% subset_iff
tff(fact_3412_subset__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ! [T: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(T),A2)
         => aa_set_set_nat_o(member_set_nat(T),B2) ) ) ).

% subset_iff
tff(fact_3413_subset__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ! [T: int] :
          ( aa_set_int_o(member_int(T),A2)
         => aa_set_int_o(member_int(T),B2) ) ) ).

% subset_iff
tff(fact_3414_subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A2)
         => aa_set_nat_o(member_nat(T),B2) ) ) ).

% subset_iff
tff(fact_3415_equalityD2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ).

% equalityD2
tff(fact_3416_equalityD1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% equalityD1
tff(fact_3417_subset__eq,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
    <=> ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),A2)
         => aa_set_complex_o(member_complex(X4),B2) ) ) ).

% subset_eq
tff(fact_3418_subset__eq,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A2)
         => aa_set_real_o(member_real(X4),B2) ) ) ).

% subset_eq
tff(fact_3419_subset__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),A2)
         => aa_set_set_nat_o(member_set_nat(X4),B2) ) ) ).

% subset_eq
tff(fact_3420_subset__eq,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => aa_set_int_o(member_int(X4),B2) ) ) ).

% subset_eq
tff(fact_3421_subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A2)
         => aa_set_nat_o(member_nat(X4),B2) ) ) ).

% subset_eq
tff(fact_3422_equalityE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% equalityE
tff(fact_3423_subsetD,axiom,
    ! [A2: set_complex,B2: set_complex,C: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
     => ( aa_set_complex_o(member_complex(C),A2)
       => aa_set_complex_o(member_complex(C),B2) ) ) ).

% subsetD
tff(fact_3424_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% subsetD
tff(fact_3425_subsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
     => ( aa_set_set_nat_o(member_set_nat(C),A2)
       => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% subsetD
tff(fact_3426_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% subsetD
tff(fact_3427_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% subsetD
tff(fact_3428_in__mono,axiom,
    ! [A2: set_complex,B2: set_complex,Xa: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => aa_set_complex_o(member_complex(Xa),B2) ) ) ).

% in_mono
tff(fact_3429_in__mono,axiom,
    ! [A2: set_real,B2: set_real,Xa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => aa_set_real_o(member_real(Xa),B2) ) ) ).

% in_mono
tff(fact_3430_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => aa_set_set_nat_o(member_set_nat(Xa),B2) ) ) ).

% in_mono
tff(fact_3431_in__mono,axiom,
    ! [A2: set_int,B2: set_int,Xa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => aa_set_int_o(member_int(Xa),B2) ) ) ).

% in_mono
tff(fact_3432_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,Xa: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => aa_set_nat_o(member_nat(Xa),B2) ) ) ).

% in_mono
tff(fact_3433_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3434_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3435_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3436_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).

% diff_right_commute
tff(fact_3437_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_3438_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% diff_right_commute
tff(fact_3439_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),C)),B) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),B)),C) ).

% diff_right_commute
tff(fact_3440_diff__commute,axiom,
    ! [Ia: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),K)),J) ).

% diff_commute
tff(fact_3441_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D4: real,Q: fun_real_o] :
      ( ! [X3: real,K2: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4))) )
     => ( ! [X3: real,K2: real] :
            ( aa_real_o(Q,X3)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4))) )
       => ! [X: real,K4: real] :
            ( ( aa_real_o(P,X)
              & aa_real_o(Q,X) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4)))
              & aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3442_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D4: rat,Q: fun_rat_o] :
      ( ! [X3: rat,K2: rat] :
          ( aa_rat_o(P,X3)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D4))) )
     => ( ! [X3: rat,K2: rat] :
            ( aa_rat_o(Q,X3)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D4))) )
       => ! [X: rat,K4: rat] :
            ( ( aa_rat_o(P,X)
              & aa_rat_o(Q,X) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4)))
              & aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3443_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D4: int,Q: fun_int_o] :
      ( ! [X3: int,K2: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D4))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D4))) )
       => ! [X: int,K4: int] :
            ( ( aa_int_o(P,X)
              & aa_int_o(Q,X) )
          <=> ( aa_int_o(P,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K4),D4)))
              & aa_int_o(Q,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3444_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D4: real,Q: fun_real_o] :
      ( ! [X3: real,K2: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4))) )
     => ( ! [X3: real,K2: real] :
            ( aa_real_o(Q,X3)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4))) )
       => ! [X: real,K4: real] :
            ( ( aa_real_o(P,X)
              | aa_real_o(Q,X) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4)))
              | aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3445_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D4: rat,Q: fun_rat_o] :
      ( ! [X3: rat,K2: rat] :
          ( aa_rat_o(P,X3)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D4))) )
     => ( ! [X3: rat,K2: rat] :
            ( aa_rat_o(Q,X3)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D4))) )
       => ! [X: rat,K4: rat] :
            ( ( aa_rat_o(P,X)
              | aa_rat_o(Q,X) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4)))
              | aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3446_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D4: int,Q: fun_int_o] :
      ( ! [X3: int,K2: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D4))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D4))) )
       => ! [X: int,K4: int] :
            ( ( aa_int_o(P,X)
              | aa_int_o(Q,X) )
          <=> ( aa_int_o(P,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K4),D4)))
              | aa_int_o(Q,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3447_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_mono
tff(fact_3448_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_mono
tff(fact_3449_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),D)) ) ) ).

% diff_mono
tff(fact_3450_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_left_mono
tff(fact_3451_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_left_mono
tff(fact_3452_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(C),A)),aa_int_int(minus_minus_int(C),B)) ) ).

% diff_left_mono
tff(fact_3453_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_right_mono
tff(fact_3454_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_right_mono
tff(fact_3455_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),C)) ) ).

% diff_right_mono
tff(fact_3456_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3457_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3458_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3459_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_3460_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3461_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3462_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(minus_minus_int(A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3463_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_strict_mono
tff(fact_3464_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_strict_mono
tff(fact_3465_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),D)) ) ) ).

% diff_strict_mono
tff(fact_3466_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_3467_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_3468_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_3469_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_strict_left_mono
tff(fact_3470_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_strict_left_mono
tff(fact_3471_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(C),A)),aa_int_int(minus_minus_int(C),B)) ) ).

% diff_strict_left_mono
tff(fact_3472_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_strict_right_mono
tff(fact_3473_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_strict_right_mono
tff(fact_3474_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),C)) ) ).

% diff_strict_right_mono
tff(fact_3475_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_3476_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_3477_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_3478_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% right_diff_distrib'
tff(fact_3479_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_3480_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_3481_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_3482_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),C)),A) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(B),A)),aa_int_int(times_times_int(C),A)) ).

% left_diff_distrib'
tff(fact_3483_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_3484_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_3485_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% right_diff_distrib
tff(fact_3486_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_3487_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_3488_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% left_diff_distrib
tff(fact_3489_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_3490_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_3491_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) = aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),aa_int_int(minus_minus_int(C),D)) ).

% add_diff_add
tff(fact_3492_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_3493_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_3494_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_3495_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% diff_diff_eq
tff(fact_3496_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_3497_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_3498_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_3499_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(plus_plus_int(C),B) = A )
     => ( C = aa_int_int(minus_minus_int(A),B) ) ) ).

% add_implies_diff
tff(fact_3500_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3501_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3502_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3503_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_3504_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_add_eq
tff(fact_3505_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),B) ).

% diff_add_eq
tff(fact_3506_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_3507_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_diff_eq2
tff(fact_3508_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),B) ).

% diff_diff_eq2
tff(fact_3509_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_3510_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) ).

% add_diff_eq
tff(fact_3511_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),C) ).

% add_diff_eq
tff(fact_3512_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3513_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3514_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(minus_minus_int(C),B) )
    <=> ( aa_int_int(plus_plus_int(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3515_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_3516_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C )
    <=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_3517_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = C )
    <=> ( A = aa_int_int(plus_plus_int(C),B) ) ) ).

% diff_eq_eq
tff(fact_3518_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3519_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3520_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(minus_minus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(minus_minus_int(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3521_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% diff_divide_distrib
tff(fact_3522_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_3523_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% diff_divide_distrib
tff(fact_3524_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(B,C)) ) ).

% dvd_diff_commute
tff(fact_3525_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(minus_minus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(minus_minus_int(B),C)) ) ).

% dvd_diff_commute
tff(fact_3526_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,Ia: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Ia)) ) ) ).

% zero_induct_lemma
tff(fact_3527_minus__nat_Odiff__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),zero_zero_nat) = Ma ).

% minus_nat.diff_0
tff(fact_3528_diffs0__imp__equal,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma) = zero_zero_nat )
       => ( Ma = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_3529_diff__less__mono2,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Ma)) ) ) ).

% diff_less_mono2
tff(fact_3530_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_3531_dvd__minus__self,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ) ).

% dvd_minus_self
tff(fact_3532_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_eq
tff(fact_3533_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_eq
tff(fact_3534_mod__diff__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B3),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A3),B3)),C) ) ) ) ).

% mod_diff_cong
tff(fact_3535_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A3,B3),C) ) ) ) ).

% mod_diff_cong
tff(fact_3536_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_left_eq
tff(fact_3537_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_left_eq
tff(fact_3538_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_right_eq
tff(fact_3539_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_right_eq
tff(fact_3540_dvd__diffD,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Ma) ) ) ) ).

% dvd_diffD
tff(fact_3541_dvd__diffD1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_3542_eq__diff__iff,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( Ma = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_3543_le__diff__iff,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ) ).

% le_diff_iff
tff(fact_3544_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_3545_diff__le__mono,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_3546_diff__le__self,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Ma) ).

% diff_le_self
tff(fact_3547_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_3548_diff__le__mono2,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Ma)) ) ).

% diff_le_mono2
tff(fact_3549_less__eq__dvd__minus,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) ) ) ).

% less_eq_dvd_minus
tff(fact_3550_diff__add__inverse2,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),Nb) = Ma ).

% diff_add_inverse2
tff(fact_3551_diff__add__inverse,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)),Nb) = Ma ).

% diff_add_inverse
tff(fact_3552_diff__cancel2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ).

% diff_cancel2
tff(fact_3553_Nat_Odiff__cancel,axiom,
    ! [K: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ).

% Nat.diff_cancel
tff(fact_3554_diff__mult__distrib,axiom,
    ! [Ma: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% diff_mult_distrib
tff(fact_3555_diff__mult__distrib2,axiom,
    ! [K: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% diff_mult_distrib2
tff(fact_3556_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = D3 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_3557_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_3558_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3559_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_3560_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_3561_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3562_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_3563_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Ta)) ) ) ).

% inf_period(3)
tff(fact_3564_inf__period_I3_J,axiom,
    ! [D: real,D4: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3565_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3566_inf__period_I3_J,axiom,
    ! [D: int,D4: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3567_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Ta)) ) ) ).

% inf_period(4)
tff(fact_3568_inf__period_I4_J,axiom,
    ! [D: real,D4: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3569_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3570_inf__period_I4_J,axiom,
    ! [D: int,D4: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3571_add__le__add__imp__diff__le,axiom,
    ! [Ia: real,K: real,Nb: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3572_add__le__add__imp__diff__le,axiom,
    ! [Ia: rat,K: rat,Nb: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Ia),K)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Ia),K)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3573_add__le__add__imp__diff__le,axiom,
    ! [Ia: nat,K: nat,Nb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3574_add__le__add__imp__diff__le,axiom,
    ! [Ia: int,K: int,Nb: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(Ia),K)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(plus_plus_int(J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(Ia),K)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(plus_plus_int(J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3575_add__le__imp__le__diff,axiom,
    ! [Ia: real,K: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ia),K)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ia),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3576_add__le__imp__le__diff,axiom,
    ! [Ia: rat,K: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Ia),K)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ia),aa_rat_rat(minus_minus_rat(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3577_add__le__imp__le__diff,axiom,
    ! [Ia: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3578_add__le__imp__le__diff,axiom,
    ! [Ia: int,K: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(Ia),K)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),aa_int_int(minus_minus_int(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3579_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_le_eq
tff(fact_3580_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_le_eq
tff(fact_3581_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(plus_plus_int(C),B)) ) ).

% diff_le_eq
tff(fact_3582_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% le_diff_eq
tff(fact_3583_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% le_diff_eq
tff(fact_3584_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(minus_minus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),C) ) ).

% le_diff_eq
tff(fact_3585_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_3586_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_3587_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3588_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3589_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3590_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3591_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3592_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3593_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3594_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3595_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3596_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3597_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3598_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(plus_plus_int(B),aa_int_int(minus_minus_int(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3599_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_less_eq
tff(fact_3600_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_less_eq
tff(fact_3601_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(plus_plus_int(C),B)) ) ).

% diff_less_eq
tff(fact_3602_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% less_diff_eq
tff(fact_3603_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% less_diff_eq
tff(fact_3604_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(minus_minus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),C) ) ).

% less_diff_eq
tff(fact_3605_square__diff__square__factored,axiom,
    ! [Xa: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y)) ).

% square_diff_square_factored
tff(fact_3606_square__diff__square__factored,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xa),Xa)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xa),Y)),aa_rat_rat(minus_minus_rat(Xa),Y)) ).

% square_diff_square_factored
tff(fact_3607_square__diff__square__factored,axiom,
    ! [Xa: int,Y: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xa),Xa)),aa_int_int(times_times_int(Y),Y)) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(Xa),Y)),aa_int_int(minus_minus_int(Xa),Y)) ).

% square_diff_square_factored
tff(fact_3608_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3609_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3610_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E2)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E2)),D) )
    <=> ( C = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3611_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3612_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3613_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E2)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E2)),D) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3614_mult__diff__mult,axiom,
    ! [Xa: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),A)),B)) ).

% mult_diff_mult
tff(fact_3615_mult__diff__mult,axiom,
    ! [Xa: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xa),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xa),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xa),A)),B)) ).

% mult_diff_mult
tff(fact_3616_mult__diff__mult,axiom,
    ! [Xa: int,Y: int,A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xa),Y)),aa_int_int(times_times_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xa),aa_int_int(minus_minus_int(Y),B))),aa_int_int(times_times_int(aa_int_int(minus_minus_int(Xa),A)),B)) ).

% mult_diff_mult
tff(fact_3617_diff__less__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_nat_nat(suc,Ma)) ).

% diff_less_Suc
tff(fact_3618_Suc__diff__Suc,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_3619_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(minus_minus_int(A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_3620_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B)) ) ).

% mod_eq_dvd_iff
tff(fact_3621_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_3622_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B))) ).

% dvd_minus_mod
tff(fact_3623_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B))) ).

% dvd_minus_mod
tff(fact_3624_diff__less,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Ma) ) ) ).

% diff_less
tff(fact_3625_Suc__diff__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) ) ) ).

% Suc_diff_le
tff(fact_3626_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_3627_less__diff__iff,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ) ).

% less_diff_iff
tff(fact_3628_diff__add__0,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = zero_zero_nat ).

% diff_add_0
tff(fact_3629_less__diff__conv,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),J) ) ).

% less_diff_conv
tff(fact_3630_add__diff__inverse__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = Ma ) ) ).

% add_diff_inverse_nat
tff(fact_3631_le__diff__conv,axiom,
    ! [J: nat,K: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),Ia)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)) ) ).

% le_diff_conv
tff(fact_3632_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_3633_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_3634_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ia)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),Ia) ) ) ).

% Nat.diff_add_assoc2
tff(fact_3635_Nat_Ole__imp__diff__is__add,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Ia) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_3636_diff__Suc__eq__diff__pred,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_3637_mod__geq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_3638_mod__if,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb),Ma,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb)) ).

% mod_if
tff(fact_3639_le__mod__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_3640_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Ma: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_3641_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_3642_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3643_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3644_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E2)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3645_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3646_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3647_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E2)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3648_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_3649_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_3650_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E2)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_3651_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3652_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3653_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E2)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3654_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3655_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3656_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3657_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xa: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xa),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3658_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3659_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xa),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xa),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3660_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xa),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xa),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3661_diff__divide__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3662_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(Xa),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xa),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3663_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xa),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xa),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3664_divide__diff__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3665_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xa),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xa),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3666_square__diff__one__factored,axiom,
    ! [Xa: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xa),Xa)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xa),one_one_complex)),aa_complex_complex(minus_minus_complex(Xa),one_one_complex)) ).

% square_diff_one_factored
tff(fact_3667_square__diff__one__factored,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),one_one_real)) ).

% square_diff_one_factored
tff(fact_3668_square__diff__one__factored,axiom,
    ! [Xa: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xa),Xa)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xa),one_one_rat)),aa_rat_rat(minus_minus_rat(Xa),one_one_rat)) ).

% square_diff_one_factored
tff(fact_3669_square__diff__one__factored,axiom,
    ! [Xa: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xa),Xa)),one_one_int) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(Xa),one_one_int)),aa_int_int(minus_minus_int(Xa),one_one_int)) ).

% square_diff_one_factored
tff(fact_3670_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_3671_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_div_mult_eq_mod
tff(fact_3672_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_3673_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_3674_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_3675_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_3676_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_3677_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B)) ).

% minus_mod_eq_mult_div
tff(fact_3678_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_3679_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_3680_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B))) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mult_div_eq_mod
tff(fact_3681_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_3682_diff__Suc__less,axiom,
    ! [Nb: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,Ia))),Nb) ) ).

% diff_Suc_less
tff(fact_3683_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => aa_nat_o(P,D5) ) ) ) ).

% nat_diff_split
tff(fact_3684_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ aa_nat_o(P,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_3685_less__diff__conv2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),Ia)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)) ) ) ).

% less_diff_conv2
tff(fact_3686_nat__eq__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),U)),Ma) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_3687_nat__eq__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( 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(minus_minus_nat,J),Ia)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_3688_nat__le__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),U)),Ma)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_3689_nat__le__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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(minus_minus_nat,J),Ia)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_3690_nat__diff__add__eq1,axiom,
    ! [J: nat,Ia: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( 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,Ia),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),U)),Ma)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_3691_nat__diff__add__eq2,axiom,
    ! [Ia: nat,J: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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(minus_minus_nat,J),Ia)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_3692_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Ma))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Ma)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_3693_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% minf(7)
tff(fact_3694_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X) ) ).

% minf(7)
tff(fact_3695_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X) ) ).

% minf(7)
tff(fact_3696_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% minf(7)
tff(fact_3697_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% minf(7)
tff(fact_3698_minf_I7_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Ta),X) ) ).

% minf(7)
tff(fact_3699_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% minf(5)
tff(fact_3700_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Ta) ) ).

% minf(5)
tff(fact_3701_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Ta) ) ).

% minf(5)
tff(fact_3702_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% minf(5)
tff(fact_3703_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% minf(5)
tff(fact_3704_minf_I5_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Ta) ) ).

% minf(5)
tff(fact_3705_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3706_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3707_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3708_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3709_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3710_minf_I4_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3711_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3712_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3713_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3714_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3715_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3716_minf_I3_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3717_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z5)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z5)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3718_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z5)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z5)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
           => ( ( aa_rat_o(P,X)
                | aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                | aa_rat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3719_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z5)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z5)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
           => ( ( aa_num_o(P,X)
                | aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                | aa_num_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3720_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z5)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z5)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3721_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3722_minf_I2_J,axiom,
    ! [P: fun_Extended_enat_o,P3: fun_Extended_enat_o,Q: fun_Extended_enat_o,Q6: fun_Extended_enat_o] :
      ( ? [Z5: extended_enat] :
        ! [X3: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X3),Z5)
         => ( aa_Extended_enat_o(P,X3)
          <=> aa_Extended_enat_o(P3,X3) ) )
     => ( ? [Z5: extended_enat] :
          ! [X3: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X3),Z5)
           => ( aa_Extended_enat_o(Q,X3)
            <=> aa_Extended_enat_o(Q6,X3) ) )
       => ? [Z4: extended_enat] :
          ! [X: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
           => ( ( aa_Extended_enat_o(P,X)
                | aa_Extended_enat_o(Q,X) )
            <=> ( aa_Extended_enat_o(P3,X)
                | aa_Extended_enat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3723_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z5)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z5)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3724_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z5)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z5)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
           => ( ( aa_rat_o(P,X)
                & aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                & aa_rat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3725_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z5)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z5)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
           => ( ( aa_num_o(P,X)
                & aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                & aa_num_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3726_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z5)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z5)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3727_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3728_minf_I1_J,axiom,
    ! [P: fun_Extended_enat_o,P3: fun_Extended_enat_o,Q: fun_Extended_enat_o,Q6: fun_Extended_enat_o] :
      ( ? [Z5: extended_enat] :
        ! [X3: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X3),Z5)
         => ( aa_Extended_enat_o(P,X3)
          <=> aa_Extended_enat_o(P3,X3) ) )
     => ( ? [Z5: extended_enat] :
          ! [X3: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X3),Z5)
           => ( aa_Extended_enat_o(Q,X3)
            <=> aa_Extended_enat_o(Q6,X3) ) )
       => ? [Z4: extended_enat] :
          ! [X: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
           => ( ( aa_Extended_enat_o(P,X)
                & aa_Extended_enat_o(Q,X) )
            <=> ( aa_Extended_enat_o(P3,X)
                & aa_Extended_enat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3729_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% pinf(7)
tff(fact_3730_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X) ) ).

% pinf(7)
tff(fact_3731_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X) ) ).

% pinf(7)
tff(fact_3732_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% pinf(7)
tff(fact_3733_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% pinf(7)
tff(fact_3734_pinf_I7_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Ta),X) ) ).

% pinf(7)
tff(fact_3735_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% pinf(5)
tff(fact_3736_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Ta) ) ).

% pinf(5)
tff(fact_3737_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Ta) ) ).

% pinf(5)
tff(fact_3738_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% pinf(5)
tff(fact_3739_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% pinf(5)
tff(fact_3740_pinf_I5_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Ta) ) ).

% pinf(5)
tff(fact_3741_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3742_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3743_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3744_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3745_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3746_pinf_I4_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3747_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3748_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3749_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3750_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3751_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3752_pinf_I3_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3753_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z5),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z5),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3754_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
           => ( ( aa_rat_o(P,X)
                | aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                | aa_rat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3755_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
           => ( ( aa_num_o(P,X)
                | aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                | aa_num_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3756_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3757_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3758_pinf_I2_J,axiom,
    ! [P: fun_Extended_enat_o,P3: fun_Extended_enat_o,Q: fun_Extended_enat_o,Q6: fun_Extended_enat_o] :
      ( ? [Z5: extended_enat] :
        ! [X3: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z5),X3)
         => ( aa_Extended_enat_o(P,X3)
          <=> aa_Extended_enat_o(P3,X3) ) )
     => ( ? [Z5: extended_enat] :
          ! [X3: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z5),X3)
           => ( aa_Extended_enat_o(Q,X3)
            <=> aa_Extended_enat_o(Q6,X3) ) )
       => ? [Z4: extended_enat] :
          ! [X: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
           => ( ( aa_Extended_enat_o(P,X)
                | aa_Extended_enat_o(Q,X) )
            <=> ( aa_Extended_enat_o(P3,X)
                | aa_Extended_enat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3759_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z5),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z5),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3760_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
           => ( ( aa_rat_o(P,X)
                & aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                & aa_rat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3761_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
           => ( ( aa_num_o(P,X)
                & aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                & aa_num_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3762_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3763_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3764_pinf_I1_J,axiom,
    ! [P: fun_Extended_enat_o,P3: fun_Extended_enat_o,Q: fun_Extended_enat_o,Q6: fun_Extended_enat_o] :
      ( ? [Z5: extended_enat] :
        ! [X3: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z5),X3)
         => ( aa_Extended_enat_o(P,X3)
          <=> aa_Extended_enat_o(P3,X3) ) )
     => ( ? [Z5: extended_enat] :
          ! [X3: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z5),X3)
           => ( aa_Extended_enat_o(Q,X3)
            <=> aa_Extended_enat_o(Q6,X3) ) )
       => ? [Z4: extended_enat] :
          ! [X: extended_enat] :
            ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
           => ( ( aa_Extended_enat_o(P,X)
                & aa_Extended_enat_o(Q,X) )
            <=> ( aa_Extended_enat_o(P3,X)
                & aa_Extended_enat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3765_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_3766_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ( Xa != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xa != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xa != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_3767_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xa) ).

% vebt_member.simps(3)
tff(fact_3768_modulo__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_3769_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( ! [Uv2: $o] : Xa != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xa != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3770_conj__le__cong,axiom,
    ! [Xa: int,X5: int,P: $o,P3: $o] :
      ( ( Xa = X5 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_3771_imp__le__cong,axiom,
    ! [Xa: int,X5: int,P: $o,P3: $o] :
      ( ( Xa = X5 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_3772_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_3773_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xa),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3774_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xa: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_3775_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,Xa: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xa),Y)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xa),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_3776_power2__commute,axiom,
    ! [Xa: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3777_power2__commute,axiom,
    ! [Xa: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3778_power2__commute,axiom,
    ! [Xa: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Y),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3779_power2__commute,axiom,
    ! [Xa: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Y),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3780_power__diff,axiom,
    ! [A: complex,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Ma)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3781_power__diff,axiom,
    ! [A: real,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3782_power__diff,axiom,
    ! [A: rat,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Ma)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3783_power__diff,axiom,
    ! [A: nat,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3784_power__diff,axiom,
    ! [A: int,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Ma)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3785_div__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_3786_div__if,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb)) ) ).

% div_if
tff(fact_3787_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_3788_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3789_add__eq__if,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) = $ite(Ma = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_3790_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xa) ).

% vebt_member.simps(4)
tff(fact_3791_nat__less__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ia),J)),U)),Ma)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_3792_nat__less__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ia),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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(minus_minus_nat,J),Ia)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_3793_mult__eq__if,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) = $ite(Ma = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_3794_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
      <=> (Y) )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xa = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xa = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3795_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,Sb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),Sb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),Sb))),V) ) ) ) ).

% scaling_mono
tff(fact_3796_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,Sb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),Sb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),Sb))),V) ) ) ) ).

% scaling_mono
tff(fact_3797_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3798_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3799_power__diff__power__eq,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ).

% power_diff_power_eq
tff(fact_3800_power__diff__power__eq,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Ma)),aa_nat_int(power_power_int(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ).

% power_diff_power_eq
tff(fact_3801_power__eq__if,axiom,
    ! [P2: complex,Ma: nat] :
      aa_nat_complex(power_power_complex(P2),Ma) = $ite(Ma = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P2),aa_nat_complex(power_power_complex(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_3802_power__eq__if,axiom,
    ! [P2: real,Ma: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Ma) = $ite(Ma = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_3803_power__eq__if,axiom,
    ! [P2: rat,Ma: nat] :
      aa_nat_rat(power_power_rat(P2),Ma) = $ite(Ma = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_3804_power__eq__if,axiom,
    ! [P2: nat,Ma: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Ma) = $ite(Ma = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_3805_power__eq__if,axiom,
    ! [P2: int,Ma: nat] :
      aa_nat_int(power_power_int(P2),Ma) = $ite(Ma = zero_zero_nat,one_one_int,aa_int_int(times_times_int(P2),aa_nat_int(power_power_int(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_3806_power__minus__mult,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3807_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_minus_mult
tff(fact_3808_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3809_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_3810_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3811_diff__le__diff__pow,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_3812_le__div__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_3813_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_3814_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% minf(8)
tff(fact_3815_minf_I8_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Ta),X) ) ).

% minf(8)
tff(fact_3816_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% minf(8)
tff(fact_3817_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% minf(8)
tff(fact_3818_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% minf(8)
tff(fact_3819_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% minf(8)
tff(fact_3820_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% minf(6)
tff(fact_3821_minf_I6_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X),Ta) ) ).

% minf(6)
tff(fact_3822_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% minf(6)
tff(fact_3823_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% minf(6)
tff(fact_3824_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% minf(6)
tff(fact_3825_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% minf(6)
tff(fact_3826_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% pinf(8)
tff(fact_3827_pinf_I8_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Ta),X) ) ).

% pinf(8)
tff(fact_3828_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% pinf(8)
tff(fact_3829_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% pinf(8)
tff(fact_3830_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% pinf(8)
tff(fact_3831_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% pinf(8)
tff(fact_3832_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% pinf(6)
tff(fact_3833_pinf_I6_J,axiom,
    ! [Ta: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X),Ta) ) ).

% pinf(6)
tff(fact_3834_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% pinf(6)
tff(fact_3835_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z4: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% pinf(6)
tff(fact_3836_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% pinf(6)
tff(fact_3837_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% pinf(6)
tff(fact_3838_list__decode_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero_nat )
     => ~ ! [N: nat] : Xa != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_3839_power2__diff,axiom,
    ! [Xa: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xa),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(Xa),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))),Xa)),Y)) ).

% power2_diff
tff(fact_3840_power2__diff,axiom,
    ! [Xa: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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))),Xa)),Y)) ).

% power2_diff
tff(fact_3841_power2__diff,axiom,
    ! [Xa: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xa),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(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_diff
tff(fact_3842_power2__diff,axiom,
    ! [Xa: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xa),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xa)),Y)) ).

% power2_diff
tff(fact_3843_mult__exp__mod__exp__eq,axiom,
    ! [Ma: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3844_mult__exp__mod__exp__eq,axiom,
    ! [Ma: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3845_mult__exp__mod__exp__eq,axiom,
    ! [Ma: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3846_zdvd__mono,axiom,
    ! [K: int,Ma: int,Ta: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ma),Ta)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(K),Ma)),aa_int_int(times_times_int(K),Ta)) ) ) ).

% zdvd_mono
tff(fact_3847_int__power__div__base,axiom,
    ! [Ma: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K),Ma)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3848_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3849_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3850_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3851_even__mask__div__iff_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% even_mask_div_iff'
tff(fact_3852_even__mask__div__iff_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% even_mask_div_iff'
tff(fact_3853_even__mask__div__iff_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% even_mask_div_iff'
tff(fact_3854_even__mask__div__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% even_mask_div_iff
tff(fact_3855_even__mask__div__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% even_mask_div_iff
tff(fact_3856_even__mask__div__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% even_mask_div_iff
tff(fact_3857_exp__div__exp__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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(minus_minus_nat,Ma),Nb))) ).

% exp_div_exp_eq
tff(fact_3858_exp__div__exp__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_int_int(divide_divide_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ))),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,Ma),Nb))) ).

% exp_div_exp_eq
tff(fact_3859_exp__div__exp__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))) ).

% exp_div_exp_eq
tff(fact_3860_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3861_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3862_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3863_minf_I10_J,axiom,
    ! [D: code_integer,Sb: code_integer] :
    ? [Z4: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z4)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb)) ) ) ).

% minf(10)
tff(fact_3864_minf_I10_J,axiom,
    ! [D: real,Sb: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Sb))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Sb)) ) ) ).

% minf(10)
tff(fact_3865_minf_I10_J,axiom,
    ! [D: rat,Sb: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Sb))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Sb)) ) ) ).

% minf(10)
tff(fact_3866_minf_I10_J,axiom,
    ! [D: nat,Sb: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb)) ) ) ).

% minf(10)
tff(fact_3867_minf_I10_J,axiom,
    ! [D: int,Sb: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Sb))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Sb)) ) ) ).

% minf(10)
tff(fact_3868_minf_I10_J,axiom,
    ! [D: extended_enat,Sb: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),plus_p3455044024723400733d_enat(X,Sb))
      <=> ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),plus_p3455044024723400733d_enat(X,Sb)) ) ) ).

% minf(10)
tff(fact_3869_minf_I9_J,axiom,
    ! [D: code_integer,Sb: code_integer] :
    ? [Z4: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb)) ) ) ).

% minf(9)
tff(fact_3870_minf_I9_J,axiom,
    ! [D: real,Sb: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Sb))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Sb)) ) ) ).

% minf(9)
tff(fact_3871_minf_I9_J,axiom,
    ! [D: rat,Sb: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Sb))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Sb)) ) ) ).

% minf(9)
tff(fact_3872_minf_I9_J,axiom,
    ! [D: nat,Sb: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb)) ) ) ).

% minf(9)
tff(fact_3873_minf_I9_J,axiom,
    ! [D: int,Sb: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Sb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Sb)) ) ) ).

% minf(9)
tff(fact_3874_minf_I9_J,axiom,
    ! [D: extended_enat,Sb: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),Z4)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),plus_p3455044024723400733d_enat(X,Sb))
      <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),plus_p3455044024723400733d_enat(X,Sb)) ) ) ).

% minf(9)
tff(fact_3875_pinf_I10_J,axiom,
    ! [D: code_integer,Sb: code_integer] :
    ? [Z4: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z4),X)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb)) ) ) ).

% pinf(10)
tff(fact_3876_pinf_I10_J,axiom,
    ! [D: real,Sb: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Sb))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Sb)) ) ) ).

% pinf(10)
tff(fact_3877_pinf_I10_J,axiom,
    ! [D: rat,Sb: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Sb))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Sb)) ) ) ).

% pinf(10)
tff(fact_3878_pinf_I10_J,axiom,
    ! [D: nat,Sb: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb)) ) ) ).

% pinf(10)
tff(fact_3879_pinf_I10_J,axiom,
    ! [D: int,Sb: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Sb))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Sb)) ) ) ).

% pinf(10)
tff(fact_3880_pinf_I10_J,axiom,
    ! [D: extended_enat,Sb: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),plus_p3455044024723400733d_enat(X,Sb))
      <=> ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),plus_p3455044024723400733d_enat(X,Sb)) ) ) ).

% pinf(10)
tff(fact_3881_pinf_I9_J,axiom,
    ! [D: code_integer,Sb: code_integer] :
    ? [Z4: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z4),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Sb)) ) ) ).

% pinf(9)
tff(fact_3882_pinf_I9_J,axiom,
    ! [D: real,Sb: real] :
    ? [Z4: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Sb))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Sb)) ) ) ).

% pinf(9)
tff(fact_3883_pinf_I9_J,axiom,
    ! [D: rat,Sb: rat] :
    ? [Z4: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Sb))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Sb)) ) ) ).

% pinf(9)
tff(fact_3884_pinf_I9_J,axiom,
    ! [D: nat,Sb: nat] :
    ? [Z4: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Sb)) ) ) ).

% pinf(9)
tff(fact_3885_pinf_I9_J,axiom,
    ! [D: int,Sb: int] :
    ? [Z4: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Sb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X),Sb)) ) ) ).

% pinf(9)
tff(fact_3886_pinf_I9_J,axiom,
    ! [D: extended_enat,Sb: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z4),X)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),plus_p3455044024723400733d_enat(X,Sb))
      <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(dvd_dv3785147216227455552d_enat,D),plus_p3455044024723400733d_enat(X,Sb)) ) ) ).

% pinf(9)
tff(fact_3887_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mi: nat,Ma: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Xa) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3888_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( Xa = Mi )
          | ( Xa = Ma )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_3889_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mi: nat,Ma: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Xa)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xa = Mi )
          | ( Xa = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_3890_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Ma)
          & ? [M2: nat] :
              ( ( some_nat(M2) = vEBT_vebt_mint(Summarya) )
              & aa_nat_o(aa_nat_fun_nat_o(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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_3891_succ__list__to__short,axiom,
    ! [Dega: nat,Mi: nat,Xa: nat,TreeLista: list_VEBT_VEBT,Ma: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_3892_pred__list__to__short,axiom,
    ! [Dega: nat,Xa: nat,Ma: nat,TreeLista: list_VEBT_VEBT,Mi: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_3893_vebt__pred_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_3894_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu),(B)),zero_zero_nat) = $ite((B),some_nat(one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_3895_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_3896_Diff__empty,axiom,
    ! [A2: set_o] : minus_minus_set_o(A2,bot_bot_set_o) = A2 ).

% Diff_empty
tff(fact_3897_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_3898_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_3899_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_3900_empty__Diff,axiom,
    ! [A2: set_o] : minus_minus_set_o(bot_bot_set_o,A2) = bot_bot_set_o ).

% empty_Diff
tff(fact_3901_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_3902_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_3903_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_3904_Diff__cancel,axiom,
    ! [A2: set_o] : minus_minus_set_o(A2,A2) = bot_bot_set_o ).

% Diff_cancel
tff(fact_3905_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_3906_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_3907_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_3908_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_3909_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mi = Ma )
       => ( ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) )
          & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_3910_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_3911_insert__simp__mima,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        | ( Xa = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_3912_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( minus_minus_set_real(A2,B2) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_3913_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( minus_minus_set_o(A2,B2) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_3914_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_3915_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat(A2,B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_3916_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_3917_succ__min,axiom,
    ! [Dega: nat,Xa: nat,Mi: nat,Ma: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = some_nat(Mi) ) ) ) ).

% succ_min
tff(fact_3918_pred__max,axiom,
    ! [Dega: nat,Ma: nat,Xa: nat,Mi: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_3919_Diff__mono,axiom,
    ! [A2: set_nat,C5: set_nat,D4: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D4),B2)
       => aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,B2)),minus_minus_set_nat(C5,D4)) ) ) ).

% Diff_mono
tff(fact_3920_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,B2)),A2) ).

% Diff_subset
tff(fact_3921_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C5)
       => ( minus_minus_set_nat(B2,minus_minus_set_nat(C5,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_3922_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(minus_minus_int(K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_3923_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z2: int] : aa_int_int(times_times_int(W),aa_int_int(minus_minus_int(Z1),Z2)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(W),Z1)),aa_int_int(times_times_int(W),Z2)) ).

% int_distrib(4)
tff(fact_3924_int__distrib_I3_J,axiom,
    ! [Z1: int,Z2: int,W: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(Z1),Z2)),W) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(Z1),W)),aa_int_int(times_times_int(Z2),W)) ).

% int_distrib(3)
tff(fact_3925_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A2),B2)
     => ? [B4: complex] : aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_3926_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
     => ? [B4: real] : aa_set_real_o(member_real(B4),minus_minus_set_real(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_3927_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B2)
     => ? [B4: set_nat] : aa_set_set_nat_o(member_set_nat(B4),minus_2163939370556025621et_nat(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_3928_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
     => ? [B4: int] : aa_set_int_o(member_int(B4),minus_minus_set_int(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_3929_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ? [B4: nat] : aa_set_nat_o(member_nat(B4),minus_minus_set_nat(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_3930_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(K),L)) ).

% signed_take_bit_diff
tff(fact_3931_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xa: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Z),Y)
     => ( plus_p3455044024723400733d_enat(Xa,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xa,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_3932_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P1,X3)
          <=> aa_int_o(P1,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_12: int] : aa_int_o(P1,X_12)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_3933_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P3,X3)
          <=> aa_int_o(P3,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P3,X3) ) )
         => ( ? [X_12: int] : aa_int_o(P3,X_12)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_3934_vebt__mint_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ! [A4: $o,B4: $o] : Xa != vEBT_Leaf((A4),(B4))
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_3935_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb)),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3936_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = some_nat(Mi) ).

% vebt_mint.simps(3)
tff(fact_3937_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = some_nat(Ma) ).

% vebt_maxt.simps(3)
tff(fact_3938_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(minus_minus_int(X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X: int] :
              ( aa_int_o(P,X)
             => aa_int_o(P,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3939_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_3940_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_3941_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(minus_minus_int(K),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_diff_iff
tff(fact_3942_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_3943_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_3944_vebt__mint_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (A4),
                  some_nat(zero_zero_nat),
                  $ite((B4),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3945_vebt__maxt_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (B4),
                  some_nat(one_one_nat),
                  $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3946_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_3947_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xa),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X4) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_3948_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xa),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_3949_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_3950_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_3951_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_3952_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv)),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_3953_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va2),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_3954_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz),Va2) = none_nat ).

% vebt_succ.simps(3)
tff(fact_3955_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Ma: nat,Dega: nat,Mi: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Ma)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
         => ( ( Ma = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))
                   => ( ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_1)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mi = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mi != Maa )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_3956_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Ma: nat,Dega: nat,Mi: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Ma)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
         => ( ( Ma = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))
                   => ( ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_1)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mi = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mi != Maa )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_3957_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_3958_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_3959_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A4: $o,B4: $o] : A1 = vEBT_Leaf((A4),(B4))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M2: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary2) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                     => aa_nat_o(vEBT_invar_vebt(X),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary2),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] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M2: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary2) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                       => aa_nat_o(vEBT_invar_vebt(X),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary2),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] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList2,Summary2) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(vEBT_invar_vebt(X),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary2),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] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                   => ( ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_1)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I3: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I3 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                                                & ! [X: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I3 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList2,Summary2) )
                   => ( ( A22 = Deg )
                     => ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(vEBT_invar_vebt(X),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary2),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] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                     => ( ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_1)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                                         => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_12) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I3: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I3 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                                                  & ! [X: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I3 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3960_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A5: $o,B5: $o] : A1 = vEBT_Leaf((A5),(B5))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_1)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_1) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_1)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_1) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
               => ( ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I)),X_1)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_1) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,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 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)))
               => ( ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I)),X_1)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_1) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,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 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3961_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A),(Uw)),aa_nat_nat(suc,zero_zero_nat)) = $ite((A),some_nat(zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_3962_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_3963_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_3964_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Va2: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Mi)
         => ( ( Ma != Mi )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_3965_divmod__step__eq,axiom,
    ! [L: num,Q2: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_3966_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R2: int] :
      unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int),aa_int_int(minus_minus_int(R2),aa_num_int(numeral_numeral_int,L))),product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_3967_divmod__step__eq,axiom,
    ! [L: num,Q2: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_3968_del__single__cont,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & ( Xa = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya) ) ) ) ).

% del_single_cont
tff(fact_3969_delt__out__of__range,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya) ) ) ) ).

% delt_out_of_range
tff(fact_3970_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Ta)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_3971_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mi != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_3972_delete__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(Ta,Xa)),Nb) ) ).

% delete_pres_valid
tff(fact_3973_dele__bmo__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_delete(Ta,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_3974_dele__member__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_delete(Ta,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa_nat_o(vEBT_vebt_member(Ta),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_3975_DiffI,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),A2)
     => ( ~ aa_set_complex_o(member_complex(C),B2)
       => aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A2,B2)) ) ) ).

% DiffI
tff(fact_3976_DiffI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),A2)
     => ( ~ aa_set_real_o(member_real(C),B2)
       => aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2)) ) ) ).

% DiffI
tff(fact_3977_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(C),B2)
       => aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2)) ) ) ).

% DiffI
tff(fact_3978_DiffI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => ( ~ aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2)) ) ) ).

% DiffI
tff(fact_3979_DiffI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => ( ~ aa_set_nat_o(member_nat(C),B2)
       => aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2)) ) ) ).

% DiffI
tff(fact_3980_Diff__iff,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A2,B2))
    <=> ( aa_set_complex_o(member_complex(C),A2)
        & ~ aa_set_complex_o(member_complex(C),B2) ) ) ).

% Diff_iff
tff(fact_3981_Diff__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
    <=> ( aa_set_real_o(member_real(C),A2)
        & ~ aa_set_real_o(member_real(C),B2) ) ) ).

% Diff_iff
tff(fact_3982_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))
    <=> ( aa_set_set_nat_o(member_set_nat(C),A2)
        & ~ aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% Diff_iff
tff(fact_3983_Diff__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        & ~ aa_set_int_o(member_int(C),B2) ) ) ).

% Diff_iff
tff(fact_3984_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        & ~ aa_set_nat_o(member_nat(C),B2) ) ) ).

% Diff_iff
tff(fact_3985_Diff__idemp,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A2,B2),B2) = minus_minus_set_nat(A2,B2) ).

% Diff_idemp
tff(fact_3986_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or8904488021354931149Most_o((A),(B)) )
    <=> ~ aa_o_o(ord_less_eq_o((A)),(B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3987_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3988_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3989_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3990_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3991_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3992_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3993_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(ord_less_eq_o((A)),(B)) ) ).

% atLeastatMost_empty_iff
tff(fact_3994_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3995_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3996_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3997_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3998_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3999_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4000_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(ord_less_o((B)),(A))
     => ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastatMost_empty
tff(fact_4001_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_4002_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_4003_atLeastatMost__empty,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( set_or5403411693681687835d_enat(A,B) = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
tff(fact_4004_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4005_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4006_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4007_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_4008_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_4009_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_4010_DiffE,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A2,B2))
     => ~ ( aa_set_complex_o(member_complex(C),A2)
         => aa_set_complex_o(member_complex(C),B2) ) ) ).

% DiffE
tff(fact_4011_DiffE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => ~ ( aa_set_real_o(member_real(C),A2)
         => aa_set_real_o(member_real(C),B2) ) ) ).

% DiffE
tff(fact_4012_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))
     => ~ ( aa_set_set_nat_o(member_set_nat(C),A2)
         => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% DiffE
tff(fact_4013_DiffE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => ~ ( aa_set_int_o(member_int(C),A2)
         => aa_set_int_o(member_int(C),B2) ) ) ).

% DiffE
tff(fact_4014_DiffE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2))
     => ~ ( aa_set_nat_o(member_nat(C),A2)
         => aa_set_nat_o(member_nat(C),B2) ) ) ).

% DiffE
tff(fact_4015_DiffD1,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A2,B2))
     => aa_set_complex_o(member_complex(C),A2) ) ).

% DiffD1
tff(fact_4016_DiffD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => aa_set_real_o(member_real(C),A2) ) ).

% DiffD1
tff(fact_4017_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))
     => aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% DiffD1
tff(fact_4018_DiffD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => aa_set_int_o(member_int(C),A2) ) ).

% DiffD1
tff(fact_4019_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2))
     => aa_set_nat_o(member_nat(C),A2) ) ).

% DiffD1
tff(fact_4020_DiffD2,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),minus_811609699411566653omplex(A2,B2))
     => ~ aa_set_complex_o(member_complex(C),B2) ) ).

% DiffD2
tff(fact_4021_DiffD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),minus_minus_set_real(A2,B2))
     => ~ aa_set_real_o(member_real(C),B2) ) ).

% DiffD2
tff(fact_4022_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),minus_2163939370556025621et_nat(A2,B2))
     => ~ aa_set_set_nat_o(member_set_nat(C),B2) ) ).

% DiffD2
tff(fact_4023_DiffD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A2,B2))
     => ~ aa_set_int_o(member_int(C),B2) ) ).

% DiffD2
tff(fact_4024_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B2))
     => ~ aa_set_nat_o(member_nat(C),B2) ) ).

% DiffD2
tff(fact_4025_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xa: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : Xa != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : Xa != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A4: nat,B4: nat] : Xa != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A4),some_nat(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4026_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xa: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : Xa != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : Xa != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] : Xa != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A4),some_P7363390416028606310at_nat(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4027_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xa: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : Xa != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V2: num] : Xa != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A4: num,B4: num] : Xa != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A4),some_num(B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4028_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xa: produc1615844007061411358on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_o,Uv2: option_nat] : Xa != produc7967433646104525454on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_o,V2: nat] : Xa != produc7967433646104525454on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_o,X3: nat,Y3: nat] : Xa != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(some_nat(X3),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4029_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xa: produc3739950874144486240at_nat] :
      ( ! [Uu2: fun_Pr2028404973800837934_nat_o,Uv2: option4927543243414619207at_nat] : Xa != produc3653971535690468304at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : Xa != produc3653971535690468304at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2028404973800837934_nat_o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : Xa != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X3),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4030_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xa: produc50614890452615238on_num] :
      ( ! [Uu2: fun_num_fun_num_o,Uv2: option_num] : Xa != produc1140783415258738358on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_o,V2: num] : Xa != produc1140783415258738358on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_o,X3: num,Y3: num] : Xa != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(some_num(X3),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4031_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_4032_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_4033_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_4034_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),Nb)
          & aa_nat_o(P,M3) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less
tff(fact_4035_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),Nb)
         => aa_nat_o(P,M3) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less
tff(fact_4036_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X3)
     => ~ ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Info2,Deg,TreeList2,Summary2),X3) ) ).

% VEBT_internal.insert'.cases
tff(fact_4037_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4038_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4039_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4040_atLeastatMost__psubset__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( aa_set3337841544853333262enat_o(ord_le2529575680413868914d_enat(set_or5403411693681687835d_enat(A,B)),set_or5403411693681687835d_enat(C,D))
    <=> ( ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
          | ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),A)
            & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),D)
            & ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A)
              | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),D) ) ) )
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4041_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B)),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
          | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
            & aa_set_nat_o(ord_less_eq_set_nat(B),D)
            & ( aa_set_nat_o(ord_less_set_nat(C),A)
              | aa_set_nat_o(ord_less_set_nat(B),D) ) ) )
        & aa_set_nat_o(ord_less_eq_set_nat(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4042_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4043_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4044_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4045_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4046_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4047_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X3)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_4048_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),X3)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),X3)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_4049_vebt__member_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X3)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X3)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : Xa != 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)),TreeList2,Summary2),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_4050_vebt__insert_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X3)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),X3)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),X3)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : Xa != 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)),TreeList2,Summary2),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_4051_vebt__pred_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)
     => ( ! [A4: $o,Uw2: $o] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A4: $o,B4: $o,Va: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Vf2)
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : Xa != 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)),TreeList2,Summary2),X3) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_4052_vebt__succ_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B4: $o] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B4)),zero_zero_nat)
     => ( ! [Uv2: $o,Uw2: $o,N: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve2)
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : Xa != 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)),TreeList2,Summary2),X3) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_4053_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_4054_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                $ite(Xa = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_4055_vebt__delete_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),zero_zero_nat)
     => ( ! [A4: $o,B4: $o] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A4: $o,B4: $o,N: nat] : Xa != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,aa_nat_nat(suc,N)))
         => ( ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] : Xa != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary2),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X3: nat] : Xa != 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] : Xa != 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : Xa != 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)),TreeList2,Summary2),X3) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_4056_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,Xa: 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),Xa) = 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_4057_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,Xa: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2,Smry2),Xa) = 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_4058_delete__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,Xa)) = minus_minus_set_nat(vEBT_set_vebt(Ta),insert_nat(Xa,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_4059_succ__less__length__list,axiom,
    ! [Dega: nat,Mi: nat,Xa: nat,TreeLista: list_VEBT_VEBT,Ma: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_4060_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_4061_insert__absorb2,axiom,
    ! [Xa: nat,A2: set_nat] : insert_nat(Xa,insert_nat(Xa,A2)) = insert_nat(Xa,A2) ).

% insert_absorb2
tff(fact_4062_insert__absorb2,axiom,
    ! [Xa: int,A2: set_int] : insert_int(Xa,insert_int(Xa,A2)) = insert_int(Xa,A2) ).

% insert_absorb2
tff(fact_4063_insert__absorb2,axiom,
    ! [Xa: real,A2: set_real] : insert_real(Xa,insert_real(Xa,A2)) = insert_real(Xa,A2) ).

% insert_absorb2
tff(fact_4064_insert__absorb2,axiom,
    ! [Xa: $o,A2: set_o] : insert_o((Xa),insert_o((Xa),A2)) = insert_o((Xa),A2) ).

% insert_absorb2
tff(fact_4065_insert__iff,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B),A2))
    <=> ( ( (A)
        <=> (B) )
        | aa_set_o_o(member_o((A)),A2) ) ) ).

% insert_iff
tff(fact_4066_insert__iff,axiom,
    ! [A: complex,B: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(A),insert_complex(B,A2))
    <=> ( ( A = B )
        | aa_set_complex_o(member_complex(A),A2) ) ) ).

% insert_iff
tff(fact_4067_insert__iff,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B,A2))
    <=> ( ( A = B )
        | aa_set_real_o(member_real(A),A2) ) ) ).

% insert_iff
tff(fact_4068_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,A2))
    <=> ( ( A = B )
        | aa_set_set_nat_o(member_set_nat(A),A2) ) ) ).

% insert_iff
tff(fact_4069_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A2))
    <=> ( ( A = B )
        | aa_set_nat_o(member_nat(A),A2) ) ) ).

% insert_iff
tff(fact_4070_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A2))
    <=> ( ( A = B )
        | aa_set_int_o(member_int(A),A2) ) ) ).

% insert_iff
tff(fact_4071_insertCI,axiom,
    ! [A: $o,B2: set_o,B: $o] :
      ( ( ~ aa_set_o_o(member_o((A)),B2)
       => ( (A)
        <=> (B) ) )
     => aa_set_o_o(member_o((A)),insert_o((B),B2)) ) ).

% insertCI
tff(fact_4072_insertCI,axiom,
    ! [A: complex,B2: set_complex,B: complex] :
      ( ( ~ aa_set_complex_o(member_complex(A),B2)
       => ( A = B ) )
     => aa_set_complex_o(member_complex(A),insert_complex(B,B2)) ) ).

% insertCI
tff(fact_4073_insertCI,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( ~ aa_set_real_o(member_real(A),B2)
       => ( A = B ) )
     => aa_set_real_o(member_real(A),insert_real(B,B2)) ) ).

% insertCI
tff(fact_4074_insertCI,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
       => ( A = B ) )
     => aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,B2)) ) ).

% insertCI
tff(fact_4075_insertCI,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A),B2)
       => ( A = B ) )
     => aa_set_nat_o(member_nat(A),insert_nat(B,B2)) ) ).

% insertCI
tff(fact_4076_insertCI,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( ~ aa_set_int_o(member_int(A),B2)
       => ( A = B ) )
     => aa_set_int_o(member_int(A),insert_int(B,B2)) ) ).

% insertCI
tff(fact_4077_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xa: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xa),Ia,Y) = list_u1324408373059187874T_VEBT(Xs,Ia,Y) ).

% list_update_overwrite
tff(fact_4078_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xa) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_af(Ta),Xa)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_4079_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xa) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ag(Ta),Xa)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_4080_delete__correct_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,Xa)) = minus_minus_set_nat(vEBT_VEBT_set_vebt(Ta),insert_nat(Xa,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_4081_singletonI,axiom,
    ! [A: complex] : aa_set_complex_o(member_complex(A),insert_complex(A,bot_bot_set_complex)) ).

% singletonI
tff(fact_4082_singletonI,axiom,
    ! [A: set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,bot_bot_set_set_nat)) ).

% singletonI
tff(fact_4083_singletonI,axiom,
    ! [A: real] : aa_set_real_o(member_real(A),insert_real(A,bot_bot_set_real)) ).

% singletonI
tff(fact_4084_singletonI,axiom,
    ! [A: $o] : aa_set_o_o(member_o((A)),insert_o((A),bot_bot_set_o)) ).

% singletonI
tff(fact_4085_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat(A),insert_nat(A,bot_bot_set_nat)) ).

% singletonI
tff(fact_4086_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int(A),insert_int(A,bot_bot_set_int)) ).

% singletonI
tff(fact_4087_insert__subset,axiom,
    ! [Xa: $o,A2: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(insert_o((Xa),A2)),B2)
    <=> ( aa_set_o_o(member_o((Xa)),B2)
        & aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ) ).

% insert_subset
tff(fact_4088_insert__subset,axiom,
    ! [Xa: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(insert_complex(Xa,A2)),B2)
    <=> ( aa_set_complex_o(member_complex(Xa),B2)
        & aa_set_complex_o(ord_le211207098394363844omplex(A2),B2) ) ) ).

% insert_subset
tff(fact_4089_insert__subset,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real(Xa,A2)),B2)
    <=> ( aa_set_real_o(member_real(Xa),B2)
        & aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ) ).

% insert_subset
tff(fact_4090_insert__subset,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(Xa,A2)),B2)
    <=> ( aa_set_set_nat_o(member_set_nat(Xa),B2)
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ) ).

% insert_subset
tff(fact_4091_insert__subset,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int(Xa,A2)),B2)
    <=> ( aa_set_int_o(member_int(Xa),B2)
        & aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ) ).

% insert_subset
tff(fact_4092_insert__subset,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(insert_nat(Xa,A2)),B2)
    <=> ( aa_set_nat_o(member_nat(Xa),B2)
        & aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ) ).

% insert_subset
tff(fact_4093_insert__Diff1,axiom,
    ! [Xa: $o,B2: set_o,A2: set_o] :
      ( aa_set_o_o(member_o((Xa)),B2)
     => ( minus_minus_set_o(insert_o((Xa),A2),B2) = minus_minus_set_o(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4094_insert__Diff1,axiom,
    ! [Xa: complex,B2: set_complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(Xa),B2)
     => ( minus_811609699411566653omplex(insert_complex(Xa,A2),B2) = minus_811609699411566653omplex(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4095_insert__Diff1,axiom,
    ! [Xa: real,B2: set_real,A2: set_real] :
      ( aa_set_real_o(member_real(Xa),B2)
     => ( minus_minus_set_real(insert_real(Xa,A2),B2) = minus_minus_set_real(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4096_insert__Diff1,axiom,
    ! [Xa: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),B2)
     => ( minus_2163939370556025621et_nat(insert_set_nat(Xa,A2),B2) = minus_2163939370556025621et_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4097_insert__Diff1,axiom,
    ! [Xa: int,B2: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(Xa),B2)
     => ( minus_minus_set_int(insert_int(Xa,A2),B2) = minus_minus_set_int(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4098_insert__Diff1,axiom,
    ! [Xa: nat,B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),B2)
     => ( minus_minus_set_nat(insert_nat(Xa,A2),B2) = minus_minus_set_nat(A2,B2) ) ) ).

% insert_Diff1
tff(fact_4099_Diff__insert0,axiom,
    ! [Xa: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A2)
     => ( minus_minus_set_o(A2,insert_o((Xa),B2)) = minus_minus_set_o(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4100_Diff__insert0,axiom,
    ! [Xa: complex,A2: set_complex,B2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xa),A2)
     => ( minus_811609699411566653omplex(A2,insert_complex(Xa,B2)) = minus_811609699411566653omplex(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4101_Diff__insert0,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( minus_minus_set_real(A2,insert_real(Xa,B2)) = minus_minus_set_real(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4102_Diff__insert0,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,B2)) = minus_2163939370556025621et_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4103_Diff__insert0,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( minus_minus_set_int(A2,insert_int(Xa,B2)) = minus_minus_set_int(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4104_Diff__insert0,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( minus_minus_set_nat(A2,insert_nat(Xa,B2)) = minus_minus_set_nat(A2,B2) ) ) ).

% Diff_insert0
tff(fact_4105_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xa: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xa)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_4106_length__list__update,axiom,
    ! [Xs: list_o,Ia: nat,Xa: $o] : size_size_list_o(list_update_o(Xs,Ia,(Xa))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_4107_length__list__update,axiom,
    ! [Xs: list_nat,Ia: nat,Xa: nat] : size_size_list_nat(list_update_nat(Xs,Ia,Xa)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_4108_length__list__update,axiom,
    ! [Xs: list_int,Ia: nat,Xa: int] : size_size_list_int(list_update_int(Xs,Ia,Xa)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_4109_nth__list__update__neq,axiom,
    ! [Ia: nat,J: nat,Xs: list_int,Xa: int] :
      ( ( Ia != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,Ia,Xa)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_4110_nth__list__update__neq,axiom,
    ! [Ia: nat,J: nat,Xs: list_nat,Xa: nat] :
      ( ( Ia != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,Ia,Xa)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_4111_nth__list__update__neq,axiom,
    ! [Ia: nat,J: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( ( Ia != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xa),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_4112_list__update__id,axiom,
    ! [Xs: list_int,Ia: nat] : list_update_int(Xs,Ia,aa_nat_int(nth_int(Xs),Ia)) = Xs ).

% list_update_id
tff(fact_4113_list__update__id,axiom,
    ! [Xs: list_nat,Ia: nat] : list_update_nat(Xs,Ia,aa_nat_nat(nth_nat(Xs),Ia)) = Xs ).

% list_update_id
tff(fact_4114_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat] : list_u1324408373059187874T_VEBT(Xs,Ia,nth_VEBT_VEBT(Xs,Ia)) = Xs ).

% list_update_id
tff(fact_4115_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_ah(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_4116_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aTP_Lamm_ai(A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_4117_singleton__conv,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_aj,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv
tff(fact_4118_singleton__conv,axiom,
    ! [A: $o] : collect_o(aTP_Lamm_ak((A))) = insert_o((A),bot_bot_set_o) ).

% singleton_conv
tff(fact_4119_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_al,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_4120_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_am,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_4121_singleton__conv2,axiom,
    ! [A: list_nat] : collect_list_nat(fequal_list_nat(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv2
tff(fact_4122_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(fequal_set_nat(A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_4123_singleton__conv2,axiom,
    ! [A: real] : collect_real(aa_real_fun_real_o(fequal_real,A)) = insert_real(A,bot_bot_set_real) ).

% singleton_conv2
tff(fact_4124_singleton__conv2,axiom,
    ! [A: $o] : collect_o(fequal_o((A))) = insert_o((A),bot_bot_set_o) ).

% singleton_conv2
tff(fact_4125_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_4126_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(fequal_int,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_4127_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( insert_real(A,A2) = insert_real(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4128_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B: $o] :
      ( ( insert_o((A),A2) = insert_o((B),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4129_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( insert_int(A,A2) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4130_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( insert_nat(A,A2) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4131_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(A,A2) )
    <=> ( ( A = B )
        & aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4132_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A2: set_o] :
      ( ( insert_o((B),bot_bot_set_o) = insert_o((A),A2) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4133_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A2) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4134_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A2) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_4135_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = insert_o((C),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & ( (B)
        <=> (C) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4136_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = insert_nat(C,bot_bot_set_nat) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4137_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( set_or1266510415728281911st_int(A,B) = insert_int(C,bot_bot_set_int) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4138_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( set_or1222579329274155063t_real(A,B) = insert_real(C,bot_bot_set_real) )
    <=> ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4139_atLeastAtMost__singleton,axiom,
    ! [A: $o] : set_or8904488021354931149Most_o((A),(A)) = insert_o((A),bot_bot_set_o) ).

% atLeastAtMost_singleton
tff(fact_4140_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_4141_atLeastAtMost__singleton,axiom,
    ! [A: int] : set_or1266510415728281911st_int(A,A) = insert_int(A,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_4142_atLeastAtMost__singleton,axiom,
    ! [A: real] : set_or1222579329274155063t_real(A,A) = insert_real(A,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_4143_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).

% insert_Diff_single
tff(fact_4144_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = insert_o((A),A2) ).

% insert_Diff_single
tff(fact_4145_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] : insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).

% insert_Diff_single
tff(fact_4146_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).

% insert_Diff_single
tff(fact_4147_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),Ia)
     => ( list_u1324408373059187874T_VEBT(Xs,Ia,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_4148_list__update__beyond,axiom,
    ! [Xs: list_o,Ia: nat,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),Ia)
     => ( list_update_o(Xs,Ia,(Xa)) = Xs ) ) ).

% list_update_beyond
tff(fact_4149_list__update__beyond,axiom,
    ! [Xs: list_nat,Ia: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),Ia)
     => ( list_update_nat(Xs,Ia,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_4150_list__update__beyond,axiom,
    ! [Xs: list_int,Ia: nat,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),Ia)
     => ( list_update_int(Xs,Ia,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_4151_nth__list__update__eq,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xa),Ia) = Xa ) ) ).

% nth_list_update_eq
tff(fact_4152_nth__list__update__eq,axiom,
    ! [Ia: nat,Xs: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,Ia,(Xa))),Ia)
      <=> (Xa) ) ) ).

% nth_list_update_eq
tff(fact_4153_nth__list__update__eq,axiom,
    ! [Ia: nat,Xs: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,Ia,Xa)),Ia) = Xa ) ) ).

% nth_list_update_eq
tff(fact_4154_nth__list__update__eq,axiom,
    ! [Ia: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,Ia,Xa)),Ia) = Xa ) ) ).

% nth_list_update_eq
tff(fact_4155_set__swap,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,Ia))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_4156_set__swap,axiom,
    ! [Ia: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( set_o2(list_update_o(list_update_o(Xs,Ia,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),Ia))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_4157_set__swap,axiom,
    ! [Ia: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,Ia,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),Ia))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_4158_set__swap,axiom,
    ! [Ia: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,Ia,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),Ia))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_4159_del__x__not__mia,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Dega: nat,H: nat,L: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
               => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Ma ))),Dega,newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                $ite(Xa = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Ma))),Dega,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_4160_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Sn: vEBT_VEBT,Summarya: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                    Ma ))),Dega,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_4161_del__x__not__mi,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                   => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                    Ma ))),Dega,Newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                $ite(Xa = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_4162_del__x__mia,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $let(
                xn: nat,
                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(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,h),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Ma ))),Dega,newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                $ite(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),Dega,newlist,Summarya) ) ) ),
                    vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_4163_del__x__mi__lets__in__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Ma ))),Dega,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_4164_del__x__mi__lets__in,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Ma ))),Dega,Newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_4165_del__x__mi,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),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(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = 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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Ma ))),Dega,newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Ma))),Dega,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_4166_del__in__range,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $let(
                xn: nat,
                xn:= 
                  $ite(Xa = 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(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),Xa),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xa = Mi,xn,Mi),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,h),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,minn,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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Ma ))),Dega,newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                  $ite(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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),Dega,newlist,Summarya) ) ) ),
                      vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_4167_pred__less__length__list,axiom,
    ! [Dega: nat,Xa: nat,Ma: nat,TreeLista: list_VEBT_VEBT,Mi: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_4168_pred__lesseq__max,axiom,
    ! [Dega: nat,Xa: nat,Ma: nat,Mi: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_4169_succ__greatereq__min,axiom,
    ! [Dega: nat,Mi: nat,Xa: nat,Ma: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_4170_set__diff__eq,axiom,
    ! [A2: set_complex,B2: set_complex] : minus_811609699411566653omplex(A2,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_an(A2),B2)) ).

% set_diff_eq
tff(fact_4171_set__diff__eq,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_ao(A2),B2)) ).

% set_diff_eq
tff(fact_4172_set__diff__eq,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_ap(A2),B2)) ).

% set_diff_eq
tff(fact_4173_set__diff__eq,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_aq(A2),B2)) ).

% set_diff_eq
tff(fact_4174_set__diff__eq,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_ar(A2),B2)) ).

% set_diff_eq
tff(fact_4175_set__diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_as(A2),B2)) ).

% set_diff_eq
tff(fact_4176_minus__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] : minus_811609699411566653omplex(A2,B2) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% minus_set_def
tff(fact_4177_minus__set__def,axiom,
    ! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(minus_404528632214381484real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% minus_set_def
tff(fact_4178_minus__set__def,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% minus_set_def
tff(fact_4179_minus__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : minus_2163939370556025621et_nat(A2,B2) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% minus_set_def
tff(fact_4180_minus__set__def,axiom,
    ! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% minus_set_def
tff(fact_4181_minus__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).

% minus_set_def
tff(fact_4182_mult__commute__abs,axiom,
    ! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_at,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_4183_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_au(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_4184_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_av,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_4185_mult__commute__abs,axiom,
    ! [C: int] : aTP_Lamm_aw(C) = times_times_int(C) ).

% mult_commute_abs
tff(fact_4186_Collect__subset,axiom,
    ! [A2: set_complex,P: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ax(A2),P))),A2) ).

% Collect_subset
tff(fact_4187_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ay(A2),P))),A2) ).

% Collect_subset
tff(fact_4188_Collect__subset,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_az(A2),P)),A2) ).

% Collect_subset
tff(fact_4189_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ba(A2),P))),A2) ).

% Collect_subset
tff(fact_4190_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bb(A2),P))),A2) ).

% Collect_subset
tff(fact_4191_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bc(A2),P))),A2) ).

% Collect_subset
tff(fact_4192_less__eq__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
    <=> ord_le7644322562001619171plex_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_eq_set_def
tff(fact_4193_less__eq__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_eq_set_def
tff(fact_4194_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_eq_set_def
tff(fact_4195_less__eq__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_eq_set_def
tff(fact_4196_less__eq__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).

% less_eq_set_def
tff(fact_4197_less__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A2),B2)
    <=> ord_le4409593196786536943plex_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_4198_less__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B2)
    <=> ord_less_fun_real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_4199_less__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),B2)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_4200_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ord_less_fun_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).

% less_set_def
tff(fact_4201_less__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B2)
    <=> ord_less_fun_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_4202_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_bd) ).

% empty_def
tff(fact_4203_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_be) ).

% empty_def
tff(fact_4204_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_bf) ).

% empty_def
tff(fact_4205_empty__def,axiom,
    bot_bot_set_o = collect_o(aTP_Lamm_bg) ).

% empty_def
tff(fact_4206_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_bh) ).

% empty_def
tff(fact_4207_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_bi) ).

% empty_def
tff(fact_4208_list__update__swap,axiom,
    ! [Ia: nat,I4: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT,X5: vEBT_VEBT] :
      ( ( Ia != I4 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xa),I4,X5) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I4,X5),Ia,Xa) ) ) ).

% list_update_swap
tff(fact_4209_Collect__conv__if,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bj(A),P)) = $ite(aa_list_nat_o(P,A),insert_list_nat(A,bot_bot_set_list_nat),bot_bot_set_list_nat) ).

% Collect_conv_if
tff(fact_4210_Collect__conv__if,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bk(A),P)) = $ite(aa_set_nat_o(P,A),insert_set_nat(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if
tff(fact_4211_Collect__conv__if,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A),P)) = $ite(aa_real_o(P,A),insert_real(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if
tff(fact_4212_Collect__conv__if,axiom,
    ! [A: $o,P: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bm((A)),P)) = $ite(aa_o_o(P,(A)),insert_o((A),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if
tff(fact_4213_Collect__conv__if,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bn(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_4214_Collect__conv__if,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bo(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_4215_Collect__conv__if2,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bp(A),P)) = $ite(aa_list_nat_o(P,A),insert_list_nat(A,bot_bot_set_list_nat),bot_bot_set_list_nat) ).

% Collect_conv_if2
tff(fact_4216_Collect__conv__if2,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bq(A),P)) = $ite(aa_set_nat_o(P,A),insert_set_nat(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if2
tff(fact_4217_Collect__conv__if2,axiom,
    ! [A: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_br(A),P)) = $ite(aa_real_o(P,A),insert_real(A,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if2
tff(fact_4218_Collect__conv__if2,axiom,
    ! [A: $o,P: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bs((A)),P)) = $ite(aa_o_o(P,(A)),insert_o((A),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if2
tff(fact_4219_Collect__conv__if2,axiom,
    ! [A: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bt(A),P)) = $ite(aa_nat_o(P,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_4220_Collect__conv__if2,axiom,
    ! [A: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bu(A),P)) = $ite(aa_int_o(P,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_4221_mk__disjoint__insert,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ? [B7: set_o] :
          ( ( A2 = insert_o((A),B7) )
          & ~ aa_set_o_o(member_o((A)),B7) ) ) ).

% mk_disjoint_insert
tff(fact_4222_mk__disjoint__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(A),A2)
     => ? [B7: set_complex] :
          ( ( A2 = insert_complex(A,B7) )
          & ~ aa_set_complex_o(member_complex(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_4223_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ? [B7: set_real] :
          ( ( A2 = insert_real(A,B7) )
          & ~ aa_set_real_o(member_real(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_4224_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ? [B7: set_set_nat] :
          ( ( A2 = insert_set_nat(A,B7) )
          & ~ aa_set_set_nat_o(member_set_nat(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_4225_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ? [B7: set_nat] :
          ( ( A2 = insert_nat(A,B7) )
          & ~ aa_set_nat_o(member_nat(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_4226_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ? [B7: set_int] :
          ( ( A2 = insert_int(A,B7) )
          & ~ aa_set_int_o(member_int(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_4227_insert__commute,axiom,
    ! [Xa: nat,Y: nat,A2: set_nat] : insert_nat(Xa,insert_nat(Y,A2)) = insert_nat(Y,insert_nat(Xa,A2)) ).

% insert_commute
tff(fact_4228_insert__commute,axiom,
    ! [Xa: int,Y: int,A2: set_int] : insert_int(Xa,insert_int(Y,A2)) = insert_int(Y,insert_int(Xa,A2)) ).

% insert_commute
tff(fact_4229_insert__commute,axiom,
    ! [Xa: real,Y: real,A2: set_real] : insert_real(Xa,insert_real(Y,A2)) = insert_real(Y,insert_real(Xa,A2)) ).

% insert_commute
tff(fact_4230_insert__commute,axiom,
    ! [Xa: $o,Y: $o,A2: set_o] : insert_o((Xa),insert_o((Y),A2)) = insert_o((Y),insert_o((Xa),A2)) ).

% insert_commute
tff(fact_4231_insert__Collect,axiom,
    ! [A: $o,P: fun_o_o] : insert_o((A),collect_o(P)) = collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bv((A)),P)) ).

% insert_Collect
tff(fact_4232_insert__Collect,axiom,
    ! [A: real,P: fun_real_o] : insert_real(A,collect_real(P)) = collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bw(A),P)) ).

% insert_Collect
tff(fact_4233_insert__Collect,axiom,
    ! [A: list_nat,P: fun_list_nat_o] : insert_list_nat(A,collect_list_nat(P)) = collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bx(A),P)) ).

% insert_Collect
tff(fact_4234_insert__Collect,axiom,
    ! [A: set_nat,P: fun_set_nat_o] : insert_set_nat(A,collect_set_nat(P)) = collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_by(A),P)) ).

% insert_Collect
tff(fact_4235_insert__Collect,axiom,
    ! [A: nat,P: fun_nat_o] : insert_nat(A,collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bz(A),P)) ).

% insert_Collect
tff(fact_4236_insert__Collect,axiom,
    ! [A: int,P: fun_int_o] : insert_int(A,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ca(A),P)) ).

% insert_Collect
tff(fact_4237_insert__eq__iff,axiom,
    ! [A: $o,A2: set_o,B: $o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A2)
     => ( ~ aa_set_o_o(member_o((B)),B2)
       => ( ( insert_o((A),A2) = insert_o((B),B2) )
        <=> $ite(
              (A) = (B),
              A2 = B2,
              ? [C6: set_o] :
                ( ( A2 = insert_o((B),C6) )
                & ~ aa_set_o_o(member_o((B)),C6)
                & ( B2 = insert_o((A),C6) )
                & ~ aa_set_o_o(member_o((A)),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4238_insert__eq__iff,axiom,
    ! [A: complex,A2: set_complex,B: complex,B2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(A),A2)
     => ( ~ aa_set_complex_o(member_complex(B),B2)
       => ( ( insert_complex(A,A2) = insert_complex(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_complex] :
                ( ( A2 = insert_complex(B,C6) )
                & ~ aa_set_complex_o(member_complex(B),C6)
                & ( B2 = insert_complex(A,C6) )
                & ~ aa_set_complex_o(member_complex(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4239_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B: real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(A),A2)
     => ( ~ aa_set_real_o(member_real(B),B2)
       => ( ( insert_real(A,A2) = insert_real(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_real] :
                ( ( A2 = insert_real(B,C6) )
                & ~ aa_set_real_o(member_real(B),C6)
                & ( B2 = insert_real(A,C6) )
                & ~ aa_set_real_o(member_real(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4240_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(B),B2)
       => ( ( insert_set_nat(A,A2) = insert_set_nat(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_set_nat] :
                ( ( A2 = insert_set_nat(B,C6) )
                & ~ aa_set_set_nat_o(member_set_nat(B),C6)
                & ( B2 = insert_set_nat(A,C6) )
                & ~ aa_set_set_nat_o(member_set_nat(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4241_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A2)
     => ( ~ aa_set_nat_o(member_nat(B),B2)
       => ( ( insert_nat(A,A2) = insert_nat(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_nat] :
                ( ( A2 = insert_nat(B,C6) )
                & ~ aa_set_nat_o(member_nat(B),C6)
                & ( B2 = insert_nat(A,C6) )
                & ~ aa_set_nat_o(member_nat(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4242_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),A2)
     => ( ~ aa_set_int_o(member_int(B),B2)
       => ( ( insert_int(A,A2) = insert_int(B,B2) )
        <=> $ite(
              A = B,
              A2 = B2,
              ? [C6: set_int] :
                ( ( A2 = insert_int(B,C6) )
                & ~ aa_set_int_o(member_int(B),C6)
                & ( B2 = insert_int(A,C6) )
                & ~ aa_set_int_o(member_int(A),C6) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_4243_insert__absorb,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( insert_o((A),A2) = A2 ) ) ).

% insert_absorb
tff(fact_4244_insert__absorb,axiom,
    ! [A: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(A),A2)
     => ( insert_complex(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4245_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4246_insert__absorb,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( insert_set_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4247_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4248_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,A2) = A2 ) ) ).

% insert_absorb
tff(fact_4249_insert__ident,axiom,
    ! [Xa: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A2)
     => ( ~ aa_set_o_o(member_o((Xa)),B2)
       => ( ( insert_o((Xa),A2) = insert_o((Xa),B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4250_insert__ident,axiom,
    ! [Xa: complex,A2: set_complex,B2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xa),A2)
     => ( ~ aa_set_complex_o(member_complex(Xa),B2)
       => ( ( insert_complex(Xa,A2) = insert_complex(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4251_insert__ident,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( ~ aa_set_real_o(member_real(Xa),B2)
       => ( ( insert_real(Xa,A2) = insert_real(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4252_insert__ident,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( ~ aa_set_set_nat_o(member_set_nat(Xa),B2)
       => ( ( insert_set_nat(Xa,A2) = insert_set_nat(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4253_insert__ident,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( ~ aa_set_nat_o(member_nat(Xa),B2)
       => ( ( insert_nat(Xa,A2) = insert_nat(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4254_insert__ident,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( ~ aa_set_int_o(member_int(Xa),B2)
       => ( ( insert_int(Xa,A2) = insert_int(Xa,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_4255_insert__compr,axiom,
    ! [A: $o,B2: set_o] : insert_o((A),B2) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_cb((A)),B2)) ).

% insert_compr
tff(fact_4256_insert__compr,axiom,
    ! [A: complex,B2: set_complex] : insert_complex(A,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_cc(A),B2)) ).

% insert_compr
tff(fact_4257_insert__compr,axiom,
    ! [A: real,B2: set_real] : insert_real(A,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_cd(A),B2)) ).

% insert_compr
tff(fact_4258_insert__compr,axiom,
    ! [A: list_nat,B2: set_list_nat] : insert_list_nat(A,B2) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_ce(A),B2)) ).

% insert_compr
tff(fact_4259_insert__compr,axiom,
    ! [A: set_nat,B2: set_set_nat] : insert_set_nat(A,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_cf(A),B2)) ).

% insert_compr
tff(fact_4260_insert__compr,axiom,
    ! [A: nat,B2: set_nat] : insert_nat(A,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_cg(A),B2)) ).

% insert_compr
tff(fact_4261_insert__compr,axiom,
    ! [A: int,B2: set_int] : insert_int(A,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_ch(A),B2)) ).

% insert_compr
tff(fact_4262_Set_Oset__insert,axiom,
    ! [Xa: $o,A2: set_o] :
      ( aa_set_o_o(member_o((Xa)),A2)
     => ~ ! [B7: set_o] :
            ( ( A2 = insert_o((Xa),B7) )
           => aa_set_o_o(member_o((Xa)),B7) ) ) ).

% Set.set_insert
tff(fact_4263_Set_Oset__insert,axiom,
    ! [Xa: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(Xa),A2)
     => ~ ! [B7: set_complex] :
            ( ( A2 = insert_complex(Xa,B7) )
           => aa_set_complex_o(member_complex(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_4264_Set_Oset__insert,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_set_real_o(member_real(Xa),A2)
     => ~ ! [B7: set_real] :
            ( ( A2 = insert_real(Xa,B7) )
           => aa_set_real_o(member_real(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_4265_Set_Oset__insert,axiom,
    ! [Xa: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ~ ! [B7: set_set_nat] :
            ( ( A2 = insert_set_nat(Xa,B7) )
           => aa_set_set_nat_o(member_set_nat(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_4266_Set_Oset__insert,axiom,
    ! [Xa: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A2)
     => ~ ! [B7: set_nat] :
            ( ( A2 = insert_nat(Xa,B7) )
           => aa_set_nat_o(member_nat(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_4267_Set_Oset__insert,axiom,
    ! [Xa: int,A2: set_int] :
      ( aa_set_int_o(member_int(Xa),A2)
     => ~ ! [B7: set_int] :
            ( ( A2 = insert_int(Xa,B7) )
           => aa_set_int_o(member_int(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_4268_insertI2,axiom,
    ! [A: $o,B2: set_o,B: $o] :
      ( aa_set_o_o(member_o((A)),B2)
     => aa_set_o_o(member_o((A)),insert_o((B),B2)) ) ).

% insertI2
tff(fact_4269_insertI2,axiom,
    ! [A: complex,B2: set_complex,B: complex] :
      ( aa_set_complex_o(member_complex(A),B2)
     => aa_set_complex_o(member_complex(A),insert_complex(B,B2)) ) ).

% insertI2
tff(fact_4270_insertI2,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( aa_set_real_o(member_real(A),B2)
     => aa_set_real_o(member_real(A),insert_real(B,B2)) ) ).

% insertI2
tff(fact_4271_insertI2,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),B2)
     => aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,B2)) ) ).

% insertI2
tff(fact_4272_insertI2,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( aa_set_nat_o(member_nat(A),B2)
     => aa_set_nat_o(member_nat(A),insert_nat(B,B2)) ) ).

% insertI2
tff(fact_4273_insertI2,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( aa_set_int_o(member_int(A),B2)
     => aa_set_int_o(member_int(A),insert_int(B,B2)) ) ).

% insertI2
tff(fact_4274_insertI1,axiom,
    ! [A: $o,B2: set_o] : aa_set_o_o(member_o((A)),insert_o((A),B2)) ).

% insertI1
tff(fact_4275_insertI1,axiom,
    ! [A: complex,B2: set_complex] : aa_set_complex_o(member_complex(A),insert_complex(A,B2)) ).

% insertI1
tff(fact_4276_insertI1,axiom,
    ! [A: real,B2: set_real] : aa_set_real_o(member_real(A),insert_real(A,B2)) ).

% insertI1
tff(fact_4277_insertI1,axiom,
    ! [A: set_nat,B2: set_set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,B2)) ).

% insertI1
tff(fact_4278_insertI1,axiom,
    ! [A: nat,B2: set_nat] : aa_set_nat_o(member_nat(A),insert_nat(A,B2)) ).

% insertI1
tff(fact_4279_insertI1,axiom,
    ! [A: int,B2: set_int] : aa_set_int_o(member_int(A),insert_int(A,B2)) ).

% insertI1
tff(fact_4280_insertE,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B),A2))
     => ( ( (A)
        <=> ~ (B) )
       => aa_set_o_o(member_o((A)),A2) ) ) ).

% insertE
tff(fact_4281_insertE,axiom,
    ! [A: complex,B: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(A),insert_complex(B,A2))
     => ( ( A != B )
       => aa_set_complex_o(member_complex(A),A2) ) ) ).

% insertE
tff(fact_4282_insertE,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),insert_real(B,A2))
     => ( ( A != B )
       => aa_set_real_o(member_real(A),A2) ) ) ).

% insertE
tff(fact_4283_insertE,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,A2))
     => ( ( A != B )
       => aa_set_set_nat_o(member_set_nat(A),A2) ) ) ).

% insertE
tff(fact_4284_insertE,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A2))
     => ( ( A != B )
       => aa_set_nat_o(member_nat(A),A2) ) ) ).

% insertE
tff(fact_4285_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A2))
     => ( ( A != B )
       => aa_set_int_o(member_int(A),A2) ) ) ).

% insertE
tff(fact_4286_lambda__zero,axiom,
    aTP_Lamm_ci = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_4287_lambda__zero,axiom,
    aTP_Lamm_cj = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_4288_lambda__zero,axiom,
    aTP_Lamm_ck = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_4289_lambda__zero,axiom,
    aTP_Lamm_cl = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_4290_lambda__zero,axiom,
    aTP_Lamm_cm = times_times_int(zero_zero_int) ).

% lambda_zero
tff(fact_4291_lambda__one,axiom,
    aTP_Lamm_cn = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_4292_lambda__one,axiom,
    aTP_Lamm_co = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_4293_lambda__one,axiom,
    aTP_Lamm_cp = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_4294_lambda__one,axiom,
    aTP_Lamm_cq = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_4295_lambda__one,axiom,
    aTP_Lamm_cr = times_times_int(one_one_int) ).

% lambda_one
tff(fact_4296_set__update__subset__insert,axiom,
    ! [Xs: list_real,Ia: nat,Xa: real] : aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,Ia,Xa))),insert_real(Xa,set_real2(Xs))) ).

% set_update_subset_insert
tff(fact_4297_set__update__subset__insert,axiom,
    ! [Xs: list_o,Ia: nat,Xa: $o] : aa_set_o_o(ord_less_eq_set_o(set_o2(list_update_o(Xs,Ia,(Xa)))),insert_o((Xa),set_o2(Xs))) ).

% set_update_subset_insert
tff(fact_4298_set__update__subset__insert,axiom,
    ! [Xs: list_int,Ia: nat,Xa: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,Ia,Xa))),insert_int(Xa,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_4299_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,Ia: nat,Xa: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Ia,Xa)),insert_VEBT_VEBT(Xa,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_4300_set__update__subset__insert,axiom,
    ! [Xs: list_nat,Ia: nat,Xa: nat] : aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,Ia,Xa))),insert_nat(Xa,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_4301_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_4302_singletonD,axiom,
    ! [B: complex,A: complex] :
      ( aa_set_complex_o(member_complex(B),insert_complex(A,bot_bot_set_complex))
     => ( B = A ) ) ).

% singletonD
tff(fact_4303_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_4304_singletonD,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real(B),insert_real(A,bot_bot_set_real))
     => ( B = A ) ) ).

% singletonD
tff(fact_4305_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
     => ( (B)
      <=> (A) ) ) ).

% singletonD
tff(fact_4306_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),insert_nat(A,bot_bot_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_4307_singletonD,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_4308_singleton__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_set_complex_o(member_complex(B),insert_complex(A,bot_bot_set_complex))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_4309_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_4310_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real(B),insert_real(A,bot_bot_set_real))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_4311_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
    <=> ( (B)
      <=> (A) ) ) ).

% singleton_iff
tff(fact_4312_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),insert_nat(A,bot_bot_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_4313_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_4314_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( insert_real(A,insert_real(B,bot_bot_set_real)) = insert_real(C,insert_real(D,bot_bot_set_real)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4315_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( insert_o((A),insert_o((B),bot_bot_set_o)) = insert_o((C),insert_o((D),bot_bot_set_o)) )
    <=> ( ( ( (A)
          <=> (C) )
          & ( (B)
          <=> (D) ) )
        | ( ( (A)
          <=> (D) )
          & ( (B)
          <=> (C) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4316_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( insert_nat(A,insert_nat(B,bot_bot_set_nat)) = insert_nat(C,insert_nat(D,bot_bot_set_nat)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4317_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( insert_int(A,insert_int(B,bot_bot_set_int)) = insert_int(C,insert_int(D,bot_bot_set_int)) )
    <=> ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_4318_insert__not__empty,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) != bot_bot_set_real ).

% insert_not_empty
tff(fact_4319_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),A2) != bot_bot_set_o ).

% insert_not_empty
tff(fact_4320_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_4321_insert__not__empty,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) != bot_bot_set_int ).

% insert_not_empty
tff(fact_4322_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( insert_real(A,bot_bot_set_real) = insert_real(B,bot_bot_set_real) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_4323_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( insert_o((A),bot_bot_set_o) = insert_o((B),bot_bot_set_o) )
     => ( (A)
      <=> (B) ) ) ).

% singleton_inject
tff(fact_4324_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( insert_nat(A,bot_bot_set_nat) = insert_nat(B,bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_4325_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B,bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_4326_insert__mono,axiom,
    ! [C5: set_int,D4: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C5),D4)
     => aa_set_int_o(ord_less_eq_set_int(insert_int(A,C5)),insert_int(A,D4)) ) ).

% insert_mono
tff(fact_4327_insert__mono,axiom,
    ! [C5: set_real,D4: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(C5),D4)
     => aa_set_real_o(ord_less_eq_set_real(insert_real(A,C5)),insert_real(A,D4)) ) ).

% insert_mono
tff(fact_4328_insert__mono,axiom,
    ! [C5: set_o,D4: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(C5),D4)
     => aa_set_o_o(ord_less_eq_set_o(insert_o((A),C5)),insert_o((A),D4)) ) ).

% insert_mono
tff(fact_4329_insert__mono,axiom,
    ! [C5: set_nat,D4: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C5),D4)
     => aa_set_nat_o(ord_less_eq_set_nat(insert_nat(A,C5)),insert_nat(A,D4)) ) ).

% insert_mono
tff(fact_4330_subset__insert,axiom,
    ! [Xa: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A2)
     => ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((Xa),B2))
      <=> aa_set_o_o(ord_less_eq_set_o(A2),B2) ) ) ).

% subset_insert
tff(fact_4331_subset__insert,axiom,
    ! [Xa: complex,A2: set_complex,B2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xa),A2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),insert_complex(Xa,B2))
      <=> aa_set_complex_o(ord_le211207098394363844omplex(A2),B2) ) ) ).

% subset_insert
tff(fact_4332_subset__insert,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B2))
      <=> aa_set_real_o(ord_less_eq_set_real(A2),B2) ) ) ).

% subset_insert
tff(fact_4333_subset__insert,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(Xa,B2))
      <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2) ) ) ).

% subset_insert
tff(fact_4334_subset__insert,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B2))
      <=> aa_set_int_o(ord_less_eq_set_int(A2),B2) ) ) ).

% subset_insert
tff(fact_4335_subset__insert,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xa,B2))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ) ).

% subset_insert
tff(fact_4336_subset__insertI,axiom,
    ! [B2: set_int,A: int] : aa_set_int_o(ord_less_eq_set_int(B2),insert_int(A,B2)) ).

% subset_insertI
tff(fact_4337_subset__insertI,axiom,
    ! [B2: set_real,A: real] : aa_set_real_o(ord_less_eq_set_real(B2),insert_real(A,B2)) ).

% subset_insertI
tff(fact_4338_subset__insertI,axiom,
    ! [B2: set_o,A: $o] : aa_set_o_o(ord_less_eq_set_o(B2),insert_o((A),B2)) ).

% subset_insertI
tff(fact_4339_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : aa_set_nat_o(ord_less_eq_set_nat(B2),insert_nat(A,B2)) ).

% subset_insertI
tff(fact_4340_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(B,B2)) ) ).

% subset_insertI2
tff(fact_4341_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(B,B2)) ) ).

% subset_insertI2
tff(fact_4342_subset__insertI2,axiom,
    ! [A2: set_o,B2: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),B2)
     => aa_set_o_o(ord_less_eq_set_o(A2),insert_o((B),B2)) ) ).

% subset_insertI2
tff(fact_4343_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(B,B2)) ) ).

% subset_insertI2
tff(fact_4344_insert__Diff__if,axiom,
    ! [Xa: $o,A2: set_o,B2: set_o] :
      minus_minus_set_o(insert_o((Xa),A2),B2) = $ite(aa_set_o_o(member_o((Xa)),B2),minus_minus_set_o(A2,B2),insert_o((Xa),minus_minus_set_o(A2,B2))) ).

% insert_Diff_if
tff(fact_4345_insert__Diff__if,axiom,
    ! [Xa: complex,A2: set_complex,B2: set_complex] :
      minus_811609699411566653omplex(insert_complex(Xa,A2),B2) = $ite(aa_set_complex_o(member_complex(Xa),B2),minus_811609699411566653omplex(A2,B2),insert_complex(Xa,minus_811609699411566653omplex(A2,B2))) ).

% insert_Diff_if
tff(fact_4346_insert__Diff__if,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      minus_minus_set_real(insert_real(Xa,A2),B2) = $ite(aa_set_real_o(member_real(Xa),B2),minus_minus_set_real(A2,B2),insert_real(Xa,minus_minus_set_real(A2,B2))) ).

% insert_Diff_if
tff(fact_4347_insert__Diff__if,axiom,
    ! [Xa: set_nat,A2: set_set_nat,B2: set_set_nat] :
      minus_2163939370556025621et_nat(insert_set_nat(Xa,A2),B2) = $ite(aa_set_set_nat_o(member_set_nat(Xa),B2),minus_2163939370556025621et_nat(A2,B2),insert_set_nat(Xa,minus_2163939370556025621et_nat(A2,B2))) ).

% insert_Diff_if
tff(fact_4348_insert__Diff__if,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      minus_minus_set_int(insert_int(Xa,A2),B2) = $ite(aa_set_int_o(member_int(Xa),B2),minus_minus_set_int(A2,B2),insert_int(Xa,minus_minus_set_int(A2,B2))) ).

% insert_Diff_if
tff(fact_4349_insert__Diff__if,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      minus_minus_set_nat(insert_nat(Xa,A2),B2) = $ite(aa_set_nat_o(member_nat(Xa),B2),minus_minus_set_nat(A2,B2),insert_nat(Xa,minus_minus_set_nat(A2,B2))) ).

% insert_Diff_if
tff(fact_4350_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit0,Nb)) = $let(
        m2: extended_enat,
        m2:= aa_num_Extended_enat(numera1916890842035813515d_enat,Nb),
        plus_p3455044024723400733d_enat(m2,m2) ) ).

% numeral_code(2)
tff(fact_4351_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(m2),m2) ) ).

% numeral_code(2)
tff(fact_4352_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2) ) ).

% numeral_code(2)
tff(fact_4353_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_4354_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(plus_plus_int(m2),m2) ) ).

% numeral_code(2)
tff(fact_4355_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(m2),m2) ) ).

% numeral_code(2)
tff(fact_4356_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_4357_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).

% power_numeral_even
tff(fact_4358_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(w),w) ) ).

% power_numeral_even
tff(fact_4359_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_4360_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(times_times_int(w),w) ) ).

% power_numeral_even
tff(fact_4361_subset__singletonD,axiom,
    ! [A2: set_real,Xa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,bot_bot_set_real))
     => ( ( A2 = bot_bot_set_real )
        | ( A2 = insert_real(Xa,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_4362_subset__singletonD,axiom,
    ! [A2: set_o,Xa: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((Xa),bot_bot_set_o))
     => ( ( A2 = bot_bot_set_o )
        | ( A2 = insert_o((Xa),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_4363_subset__singletonD,axiom,
    ! [A2: set_int,Xa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,bot_bot_set_int))
     => ( ( A2 = bot_bot_set_int )
        | ( A2 = insert_int(Xa,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_4364_subset__singletonD,axiom,
    ! [A2: set_nat,Xa: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xa,bot_bot_set_nat))
     => ( ( A2 = bot_bot_set_nat )
        | ( A2 = insert_nat(Xa,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_4365_subset__singleton__iff,axiom,
    ! [X6: set_real,A: real] :
      ( aa_set_real_o(ord_less_eq_set_real(X6),insert_real(A,bot_bot_set_real))
    <=> ( ( X6 = bot_bot_set_real )
        | ( X6 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_4366_subset__singleton__iff,axiom,
    ! [X6: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(X6),insert_o((A),bot_bot_set_o))
    <=> ( ( X6 = bot_bot_set_o )
        | ( X6 = insert_o((A),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_4367_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X6),insert_int(A,bot_bot_set_int))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_4368_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X6),insert_nat(A,bot_bot_set_nat))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_4369_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( (A)
      <=> (B) )
     => ( set_or8904488021354931149Most_o((A),(B)) = insert_o((A),bot_bot_set_o) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4370_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_4371_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_4372_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_4373_Diff__insert__absorb,axiom,
    ! [Xa: complex,A2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xa),A2)
     => ( minus_811609699411566653omplex(insert_complex(Xa,A2),insert_complex(Xa,bot_bot_set_complex)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4374_Diff__insert__absorb,axiom,
    ! [Xa: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A2)
     => ( minus_2163939370556025621et_nat(insert_set_nat(Xa,A2),insert_set_nat(Xa,bot_bot_set_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4375_Diff__insert__absorb,axiom,
    ! [Xa: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A2)
     => ( minus_minus_set_real(insert_real(Xa,A2),insert_real(Xa,bot_bot_set_real)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4376_Diff__insert__absorb,axiom,
    ! [Xa: $o,A2: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A2)
     => ( minus_minus_set_o(insert_o((Xa),A2),insert_o((Xa),bot_bot_set_o)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4377_Diff__insert__absorb,axiom,
    ! [Xa: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A2)
     => ( minus_minus_set_int(insert_int(Xa,A2),insert_int(Xa,bot_bot_set_int)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4378_Diff__insert__absorb,axiom,
    ! [Xa: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A2)
     => ( minus_minus_set_nat(insert_nat(Xa,A2),insert_nat(Xa,bot_bot_set_nat)) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_4379_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B2) ).

% Diff_insert2
tff(fact_4380_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : minus_minus_set_o(A2,insert_o((A),B2)) = minus_minus_set_o(minus_minus_set_o(A2,insert_o((A),bot_bot_set_o)),B2) ).

% Diff_insert2
tff(fact_4381_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B2) ).

% Diff_insert2
tff(fact_4382_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat)),B2) ).

% Diff_insert2
tff(fact_4383_insert__Diff,axiom,
    ! [A: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(A),A2)
     => ( insert_complex(A,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = A2 ) ) ).

% insert_Diff
tff(fact_4384_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_4385_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).

% insert_Diff
tff(fact_4386_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( insert_o((A),minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = A2 ) ) ).

% insert_Diff
tff(fact_4387_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).

% insert_Diff
tff(fact_4388_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).

% insert_Diff
tff(fact_4389_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,B2),insert_real(A,bot_bot_set_real)) ).

% Diff_insert
tff(fact_4390_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : minus_minus_set_o(A2,insert_o((A),B2)) = minus_minus_set_o(minus_minus_set_o(A2,B2),insert_o((A),bot_bot_set_o)) ).

% Diff_insert
tff(fact_4391_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,B2),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_4392_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,B2),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_4393_subset__Diff__insert,axiom,
    ! [A2: set_o,B2: set_o,Xa: $o,C5: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),minus_minus_set_o(B2,insert_o((Xa),C5)))
    <=> ( aa_set_o_o(ord_less_eq_set_o(A2),minus_minus_set_o(B2,C5))
        & ~ aa_set_o_o(member_o((Xa)),A2) ) ) ).

% subset_Diff_insert
tff(fact_4394_subset__Diff__insert,axiom,
    ! [A2: set_complex,B2: set_complex,Xa: complex,C5: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),minus_811609699411566653omplex(B2,insert_complex(Xa,C5)))
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(A2),minus_811609699411566653omplex(B2,C5))
        & ~ aa_set_complex_o(member_complex(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_4395_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,Xa: real,C5: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B2,insert_real(Xa,C5)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A2),minus_minus_set_real(B2,C5))
        & ~ aa_set_real_o(member_real(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_4396_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,Xa: set_nat,C5: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,insert_set_nat(Xa,C5)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),minus_2163939370556025621et_nat(B2,C5))
        & ~ aa_set_set_nat_o(member_set_nat(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_4397_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,Xa: int,C5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B2,insert_int(Xa,C5)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A2),minus_minus_set_int(B2,C5))
        & ~ aa_set_int_o(member_int(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_4398_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,Xa: nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),minus_minus_set_nat(B2,insert_nat(Xa,C5)))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),minus_minus_set_nat(B2,C5))
        & ~ aa_set_nat_o(member_nat(Xa),A2) ) ) ).

% subset_Diff_insert
tff(fact_4399_set__update__subsetI,axiom,
    ! [Xs: list_complex,A2: set_complex,Xa: complex,Ia: nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,Ia,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_4400_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,Xa: real,Ia: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,Ia,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_4401_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,Xa: set_nat,Ia: nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A2)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A2)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,Ia,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_4402_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,Xa: int,Ia: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A2)
     => ( aa_set_int_o(member_int(Xa),A2)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,Ia,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_4403_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,Xa: vEBT_VEBT,Ia: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( member_VEBT_VEBT(Xa,A2)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Ia,Xa)),A2) ) ) ).

% set_update_subsetI
tff(fact_4404_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,Xa: nat,Ia: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A2)
     => ( aa_set_nat_o(member_nat(Xa),A2)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,Ia,Xa))),A2) ) ) ).

% set_update_subsetI
tff(fact_4405_Diff__single__insert,axiom,
    ! [A2: set_real,Xa: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B2)
     => aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B2)) ) ).

% Diff_single_insert
tff(fact_4406_Diff__single__insert,axiom,
    ! [A2: set_o,Xa: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A2,insert_o((Xa),bot_bot_set_o))),B2)
     => aa_set_o_o(ord_less_eq_set_o(A2),insert_o((Xa),B2)) ) ).

% Diff_single_insert
tff(fact_4407_Diff__single__insert,axiom,
    ! [A2: set_int,Xa: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(Xa,bot_bot_set_int))),B2)
     => aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B2)) ) ).

% Diff_single_insert
tff(fact_4408_Diff__single__insert,axiom,
    ! [A2: set_nat,Xa: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,insert_nat(Xa,bot_bot_set_nat))),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xa,B2)) ) ).

% Diff_single_insert
tff(fact_4409_subset__insert__iff,axiom,
    ! [A2: set_complex,Xa: complex,B2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),insert_complex(Xa,B2))
    <=> $ite(aa_set_complex_o(member_complex(Xa),A2),aa_set_complex_o(ord_le211207098394363844omplex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex))),B2),aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)) ) ).

% subset_insert_iff
tff(fact_4410_subset__insert__iff,axiom,
    ! [A2: set_set_nat,Xa: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),insert_set_nat(Xa,B2))
    <=> $ite(aa_set_set_nat_o(member_set_nat(Xa),A2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_4411_subset__insert__iff,axiom,
    ! [A2: set_real,Xa: real,B2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),insert_real(Xa,B2))
    <=> $ite(aa_set_real_o(member_real(Xa),A2),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A2),B2)) ) ).

% subset_insert_iff
tff(fact_4412_subset__insert__iff,axiom,
    ! [A2: set_o,Xa: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),insert_o((Xa),B2))
    <=> $ite(aa_set_o_o(member_o((Xa)),A2),aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A2,insert_o((Xa),bot_bot_set_o))),B2),aa_set_o_o(ord_less_eq_set_o(A2),B2)) ) ).

% subset_insert_iff
tff(fact_4413_subset__insert__iff,axiom,
    ! [A2: set_int,Xa: int,B2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),insert_int(Xa,B2))
    <=> $ite(aa_set_int_o(member_int(Xa),A2),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(Xa,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A2),B2)) ) ).

% subset_insert_iff
tff(fact_4414_subset__insert__iff,axiom,
    ! [A2: set_nat,Xa: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),insert_nat(Xa,B2))
    <=> $ite(aa_set_nat_o(member_nat(Xa),A2),aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,insert_nat(Xa,bot_bot_set_nat))),B2),aa_set_nat_o(ord_less_eq_set_nat(A2),B2)) ) ).

% subset_insert_iff
tff(fact_4415_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_4416_atLeastAtMost__insertL,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( insert_nat(Ma,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_4417_atLeastAtMostSuc__conv,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4418_Icc__eq__insert__lb__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb) = insert_nat(Ma,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4419_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,Xa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(Xa),set_complex2(list_update_complex(Xs,Nb,Xa))) ) ).

% set_update_memI
tff(fact_4420_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(Xa),set_real2(list_update_real(Xs,Nb,Xa))) ) ).

% set_update_memI
tff(fact_4421_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_set_nat,Xa: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(list_update_set_nat(Xs,Nb,Xa))) ) ).

% set_update_memI
tff(fact_4422_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(Xa,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xa))) ) ).

% set_update_memI
tff(fact_4423_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => aa_set_o_o(member_o((Xa)),set_o2(list_update_o(Xs,Nb,(Xa)))) ) ).

% set_update_memI
tff(fact_4424_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xa),set_nat2(list_update_nat(Xs,Nb,Xa))) ) ).

% set_update_memI
tff(fact_4425_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(Xa),set_int2(list_update_int(Xs,Nb,Xa))) ) ).

% set_update_memI
tff(fact_4426_list__update__same__conv,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,Ia,Xa) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,Ia) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_4427_list__update__same__conv,axiom,
    ! [Ia: nat,Xs: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,Ia,(Xa)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),Ia)
        <=> (Xa) ) ) ) ).

% list_update_same_conv
tff(fact_4428_list__update__same__conv,axiom,
    ! [Ia: nat,Xs: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,Ia,Xa) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),Ia) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_4429_list__update__same__conv,axiom,
    ! [Ia: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,Ia,Xa) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),Ia) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_4430_nth__list__update,axiom,
    ! [Ia: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,Ia,Xa),J) = $ite(Ia = J,Xa,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_4431_nth__list__update,axiom,
    ! [Ia: nat,Xs: list_o,Xa: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,Ia,(Xa))),J)
      <=> $ite(Ia = J,(Xa),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_4432_nth__list__update,axiom,
    ! [Ia: nat,Xs: list_nat,Xa: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,Ia,Xa)),J) = $ite(Ia = J,Xa,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_4433_nth__list__update,axiom,
    ! [Ia: nat,Xs: list_int,Xa: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,Ia,Xa)),J) = $ite(Ia = J,Xa,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_4434_psubset__insert__iff,axiom,
    ! [A2: set_complex,Xa: complex,B2: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A2),insert_complex(Xa,B2))
    <=> $ite(
          aa_set_complex_o(member_complex(Xa),B2),
          aa_set_complex_o(ord_less_set_complex(A2),B2),
          $ite(aa_set_complex_o(member_complex(Xa),A2),aa_set_complex_o(ord_less_set_complex(minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex))),B2),aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_4435_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,Xa: set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A2),insert_set_nat(Xa,B2))
    <=> $ite(
          aa_set_set_nat_o(member_set_nat(Xa),B2),
          aa_set_set_nat_o(ord_less_set_set_nat(A2),B2),
          $ite(aa_set_set_nat_o(member_set_nat(Xa),A2),aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(A2,insert_set_nat(Xa,bot_bot_set_set_nat))),B2),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_4436_psubset__insert__iff,axiom,
    ! [A2: set_real,Xa: real,B2: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),insert_real(Xa,B2))
    <=> $ite(
          aa_set_real_o(member_real(Xa),B2),
          aa_set_real_o(ord_less_set_real(A2),B2),
          $ite(aa_set_real_o(member_real(Xa),A2),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real))),B2),aa_set_real_o(ord_less_eq_set_real(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_4437_psubset__insert__iff,axiom,
    ! [A2: set_o,Xa: $o,B2: set_o] :
      ( aa_set_o_o(ord_less_set_o(A2),insert_o((Xa),B2))
    <=> $ite(
          aa_set_o_o(member_o((Xa)),B2),
          aa_set_o_o(ord_less_set_o(A2),B2),
          $ite(aa_set_o_o(member_o((Xa)),A2),aa_set_o_o(ord_less_set_o(minus_minus_set_o(A2,insert_o((Xa),bot_bot_set_o))),B2),aa_set_o_o(ord_less_eq_set_o(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_4438_psubset__insert__iff,axiom,
    ! [A2: set_int,Xa: int,B2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),insert_int(Xa,B2))
    <=> $ite(
          aa_set_int_o(member_int(Xa),B2),
          aa_set_int_o(ord_less_set_int(A2),B2),
          $ite(aa_set_int_o(member_int(Xa),A2),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A2,insert_int(Xa,bot_bot_set_int))),B2),aa_set_int_o(ord_less_eq_set_int(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_4439_psubset__insert__iff,axiom,
    ! [A2: set_nat,Xa: nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),insert_nat(Xa,B2))
    <=> $ite(
          aa_set_nat_o(member_nat(Xa),B2),
          aa_set_nat_o(ord_less_set_nat(A2),B2),
          $ite(aa_set_nat_o(member_nat(Xa),A2),aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(A2,insert_nat(Xa,bot_bot_set_nat))),B2),aa_set_nat_o(ord_less_eq_set_nat(A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_4440_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P,X3)
          <=> aa_int_o(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
       => ( ? [X_1: int] : aa_int_o(P,X_1)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4441_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X != aa_int_int(minus_minus_int(Xb2),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(plus_plus_int(X),D4)) ) ) ) ).

% aset(7)
tff(fact_4442_aset_I5_J,axiom,
    ! [D4: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Ta),A2)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X != aa_int_int(minus_minus_int(Xb2),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(X),D4)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_4443_aset_I4_J,axiom,
    ! [D4: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Ta),A2)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X != aa_int_int(minus_minus_int(Xb2),Xa3) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(plus_plus_int(X),D4) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_4444_aset_I3_J,axiom,
    ! [D4: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(plus_plus_int(Ta),one_one_int)),A2)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X != aa_int_int(minus_minus_int(Xb2),Xa3) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(plus_plus_int(X),D4) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_4445_bset_I7_J,axiom,
    ! [D4: int,Ta: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Ta),B2)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X != aa_int_int(plus_plus_int(Xb2),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(minus_minus_int(X),D4)) ) ) ) ) ).

% bset(7)
tff(fact_4446_bset_I5_J,axiom,
    ! [D4: int,B2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X != aa_int_int(plus_plus_int(Xb2),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(X),D4)),Ta) ) ) ) ).

% bset(5)
tff(fact_4447_bset_I4_J,axiom,
    ! [D4: int,Ta: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Ta),B2)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X != aa_int_int(plus_plus_int(Xb2),Xa3) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(minus_minus_int(X),D4) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_4448_bset_I3_J,axiom,
    ! [D4: int,Ta: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(minus_minus_int(Ta),one_one_int)),B2)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X != aa_int_int(plus_plus_int(Xb2),Xa3) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(minus_minus_int(X),D4) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_4449_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,Sb: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,Sb)),Xa)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_4450_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X != aa_int_int(minus_minus_int(Xb2),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(plus_plus_int(X),D4)) ) ) ) ).

% aset(8)
tff(fact_4451_aset_I6_J,axiom,
    ! [D4: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(plus_plus_int(Ta),one_one_int)),A2)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X != aa_int_int(minus_minus_int(Xb2),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(X),D4)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_4452_bset_I8_J,axiom,
    ! [D4: int,Ta: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(minus_minus_int(Ta),one_one_int)),B2)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B2)
                   => ( X != aa_int_int(plus_plus_int(Xb2),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(minus_minus_int(X),D4)) ) ) ) ) ).

% bset(8)
tff(fact_4453_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B2)
                 => ( X != aa_int_int(plus_plus_int(Xb2),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(X),D4)),Ta) ) ) ) ).

% bset(6)
tff(fact_4454_cpmi,axiom,
    ! [D4: int,P: fun_int_o,P3: fun_int_o,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),B2)
                     => ( X3 != aa_int_int(plus_plus_int(Xb3),Xa2) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(minus_minus_int(X3),D4)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D4))) )
           => ( ? [X_1: int] : aa_int_o(P,X_1)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa4: int] :
                        ( aa_set_int_o(member_int(Xa4),B2)
                        & aa_int_o(P,aa_int_int(plus_plus_int(Xa4),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4455_cppi,axiom,
    ! [D4: int,P: fun_int_o,P3: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),A2)
                     => ( X3 != aa_int_int(minus_minus_int(Xb3),Xa2) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(plus_plus_int(X3),D4)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D4))) )
           => ( ? [X_1: int] : aa_int_o(P,X_1)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa4: int] :
                        ( aa_set_int_o(member_int(Xa4),A2)
                        & aa_int_o(P,aa_int_int(minus_minus_int(Xa4),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4456_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd)),Xa)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_4457_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya)),Xa)
    <=> $ite(
          Xa = Mi,
          $true,
          $ite(
            Xa = Ma,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_4458_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeLista,Vc)),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_4459_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),Xa) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) ),
        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),
          $let(
            xn: nat,
            xn:= 
              $ite(Xa = 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(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),Xa),
            $let(
              minn: nat,
              minn:= 
                $ite(Xa = Mi,xn,Mi),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                    $let(
                      newlist: list_VEBT_VEBT,
                      newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                      $ite(
                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summarya,h),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                $ite(
                                  xn = Ma,
                                  $let(
                                    maxs: option_nat,
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                  Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,sn) ),
                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                              $ite(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,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,Summarya) ) ) ),
                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_4460_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_4461_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_4462_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_4463_vebt__delete_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != vEBT_Leaf($false,(B4)) ) ) )
       => ( ! [A4: $o] :
              ( ? [B4: $o] : Xa = vEBT_Leaf((A4),(B4))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf((A4),$false) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
                 => ( Y != vEBT_Leaf((A4),(B4)) ) ) )
           => ( ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary2) )
                 => ( Y != vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( Xa = 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] :
                      ( ( Xa = 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,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = 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(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi2,xn,Mi2),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                          $let(
                                            newlist: list_VEBT_VEBT,
                                            newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                            $ite(
                                              aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                      $ite(
                                                        xn = Ma2,
                                                        $let(
                                                          maxs: option_nat,
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none_nat,minn,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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                        Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,sn) ),
                                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                    $ite(xn = Ma2,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,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,Summary2) ) ) ),
                                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_4464_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xaa = Mi2 )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
             => ~ ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_4465_vebt__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
             => ~ $ite(
                    Xaa = Mi2,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_4466_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xa != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_4467_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_4468_vebt__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
                   => $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_4469_vebt__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_4470_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),Xa) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
        some_nat(Ma),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    pr: option_nat,
                    pr:= vEBT_vebt_pred(Summarya,h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),some_nat(Mi),none_nat),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_4471_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),Xa) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
        some_nat(Mi),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    sc: option_nat,
                    sc:= vEBT_vebt_succ(Summarya,h),
                    $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_4472_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = vEBT_Leaf((A),(B)) ).

% vebt_delete.simps(3)
tff(fact_4473_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),zero_zero_nat) = vEBT_Leaf($false,(B)) ).

% vebt_delete.simps(1)
tff(fact_4474_vebt__delete_Osimps_I4_J,axiom,
    ! [Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya),Uu) = vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya) ).

% vebt_delete.simps(4)
tff(fact_4475_vebt__pred_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] : Xa = vEBT_Leaf((A4),(Uw2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ? [Va: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ( Y != $ite(
                        (B4),
                        some_nat(one_one_nat),
                        $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              some_nat(Ma2),
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,h),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= vEBT_vebt_pred(Summary2,h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa),some_nat(Mi2),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_4476_vebt__succ_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( ! [Uu2: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((Uu2),(B4)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B4),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xa = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xa = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            some_nat(Mi2),
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,h),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= vEBT_vebt_succ(Summary2,h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_4477_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf((A),$false) ).

% vebt_delete.simps(2)
tff(fact_4478_insert__simp__excp,axiom,
    ! [Mi: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xa: nat,Ma: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_4479_insert__simp__norm,axiom,
    ! [Xa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mi: nat,Ma: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Dega,TreeLista,Summarya),Xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ma))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_4480_insert__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( sup_sup_set_nat(vEBT_set_vebt(Ta),insert_nat(Xa,bot_bot_set_nat)) = vEBT_set_vebt(vEBT_vebt_insert(Ta,Xa)) ) ) ) ).

% insert_correct
tff(fact_4481_insert__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( sup_sup_set_nat(vEBT_VEBT_set_vebt(Ta),insert_nat(Xa,bot_bot_set_nat)) = vEBT_VEBT_set_vebt(vEBT_vebt_insert(Ta,Xa)) ) ) ) ).

% insert_corr
tff(fact_4482_vebt__insert_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B4)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts2,S) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Xaa,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_4483_vebt__succ_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B4)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = $ite((B4),some_nat(one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B4)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                some_nat(Mi2),
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,h),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= vEBT_vebt_succ(Summary2,h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_4484_vebt__pred_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A4),some_nat(zero_zero_nat),none_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B4)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = $ite(
                              (B4),
                              some_nat(one_one_nat),
                              $ite((A4),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  some_nat(Ma2),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,h),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= vEBT_vebt_pred(Summary2,h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa),some_nat(Mi2),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(pr)))) ) ) ) ),
                                        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)),TreeList2,Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_4485_UnCI,axiom,
    ! [C: complex,B2: set_complex,A2: set_complex] :
      ( ( ~ aa_set_complex_o(member_complex(C),B2)
       => aa_set_complex_o(member_complex(C),A2) )
     => aa_set_complex_o(member_complex(C),sup_sup_set_complex(A2,B2)) ) ).

% UnCI
tff(fact_4486_UnCI,axiom,
    ! [C: real,B2: set_real,A2: set_real] :
      ( ( ~ aa_set_real_o(member_real(C),B2)
       => aa_set_real_o(member_real(C),A2) )
     => aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2)) ) ).

% UnCI
tff(fact_4487_UnCI,axiom,
    ! [C: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(C),B2)
       => aa_set_set_nat_o(member_set_nat(C),A2) )
     => aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2)) ) ).

% UnCI
tff(fact_4488_UnCI,axiom,
    ! [C: int,B2: set_int,A2: set_int] :
      ( ( ~ aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),A2) )
     => aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2)) ) ).

% UnCI
tff(fact_4489_UnCI,axiom,
    ! [C: nat,B2: set_nat,A2: set_nat] :
      ( ( ~ aa_set_nat_o(member_nat(C),B2)
       => aa_set_nat_o(member_nat(C),A2) )
     => aa_set_nat_o(member_nat(C),sup_sup_set_nat(A2,B2)) ) ).

% UnCI
tff(fact_4490_Un__iff,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),sup_sup_set_complex(A2,B2))
    <=> ( aa_set_complex_o(member_complex(C),A2)
        | aa_set_complex_o(member_complex(C),B2) ) ) ).

% Un_iff
tff(fact_4491_Un__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2))
    <=> ( aa_set_real_o(member_real(C),A2)
        | aa_set_real_o(member_real(C),B2) ) ) ).

% Un_iff
tff(fact_4492_Un__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2))
    <=> ( aa_set_set_nat_o(member_set_nat(C),A2)
        | aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% Un_iff
tff(fact_4493_Un__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        | aa_set_int_o(member_int(C),B2) ) ) ).

% Un_iff
tff(fact_4494_Un__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),sup_sup_set_nat(A2,B2))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        | aa_set_nat_o(member_nat(C),B2) ) ) ).

% Un_iff
tff(fact_4495_Un__empty,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( sup_sup_set_real(A2,B2) = bot_bot_set_real )
    <=> ( ( A2 = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% Un_empty
tff(fact_4496_Un__empty,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( sup_sup_set_o(A2,B2) = bot_bot_set_o )
    <=> ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% Un_empty
tff(fact_4497_Un__empty,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat(A2,B2) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_4498_Un__empty,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( sup_sup_set_int(A2,B2) = bot_bot_set_int )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_4499_Un__subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(sup_sup_set_nat(A2,B2)),C5)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),C5)
        & aa_set_nat_o(ord_less_eq_set_nat(B2),C5) ) ) ).

% Un_subset_iff
tff(fact_4500_Un__insert__left,axiom,
    ! [A: int,B2: set_int,C5: set_int] : sup_sup_set_int(insert_int(A,B2),C5) = insert_int(A,sup_sup_set_int(B2,C5)) ).

% Un_insert_left
tff(fact_4501_Un__insert__left,axiom,
    ! [A: real,B2: set_real,C5: set_real] : sup_sup_set_real(insert_real(A,B2),C5) = insert_real(A,sup_sup_set_real(B2,C5)) ).

% Un_insert_left
tff(fact_4502_Un__insert__left,axiom,
    ! [A: $o,B2: set_o,C5: set_o] : sup_sup_set_o(insert_o((A),B2),C5) = insert_o((A),sup_sup_set_o(B2,C5)) ).

% Un_insert_left
tff(fact_4503_Un__insert__left,axiom,
    ! [A: nat,B2: set_nat,C5: set_nat] : sup_sup_set_nat(insert_nat(A,B2),C5) = insert_nat(A,sup_sup_set_nat(B2,C5)) ).

% Un_insert_left
tff(fact_4504_Un__insert__right,axiom,
    ! [A2: set_int,A: int,B2: set_int] : sup_sup_set_int(A2,insert_int(A,B2)) = insert_int(A,sup_sup_set_int(A2,B2)) ).

% Un_insert_right
tff(fact_4505_Un__insert__right,axiom,
    ! [A2: set_real,A: real,B2: set_real] : sup_sup_set_real(A2,insert_real(A,B2)) = insert_real(A,sup_sup_set_real(A2,B2)) ).

% Un_insert_right
tff(fact_4506_Un__insert__right,axiom,
    ! [A2: set_o,A: $o,B2: set_o] : sup_sup_set_o(A2,insert_o((A),B2)) = insert_o((A),sup_sup_set_o(A2,B2)) ).

% Un_insert_right
tff(fact_4507_Un__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] : sup_sup_set_nat(A2,insert_nat(A,B2)) = insert_nat(A,sup_sup_set_nat(A2,B2)) ).

% Un_insert_right
tff(fact_4508_Un__Diff__cancel,axiom,
    ! [A2: set_nat,B2: set_nat] : sup_sup_set_nat(A2,minus_minus_set_nat(B2,A2)) = sup_sup_set_nat(A2,B2) ).

% Un_Diff_cancel
tff(fact_4509_Un__Diff__cancel2,axiom,
    ! [B2: set_nat,A2: set_nat] : sup_sup_set_nat(minus_minus_set_nat(B2,A2),A2) = sup_sup_set_nat(B2,A2) ).

% Un_Diff_cancel2
tff(fact_4510_max__Suc__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),Nb)) ).

% max_Suc_Suc
tff(fact_4511_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_4512_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_4513_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_4514_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_4515_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_4516_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_4517_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),numera6620942414471956472nteger(U)) ).

% max_number_of(1)
tff(fact_4518_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,U)),aa_num_Extended_enat(numera1916890842035813515d_enat,V)) = $ite(aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,U)),aa_num_Extended_enat(numera1916890842035813515d_enat,V)),aa_num_Extended_enat(numera1916890842035813515d_enat,V),aa_num_Extended_enat(numera1916890842035813515d_enat,U)) ).

% max_number_of(1)
tff(fact_4519_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(1)
tff(fact_4520_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(1)
tff(fact_4521_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_4522_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_4523_max__0__1_I3_J,axiom,
    ! [Xa: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Xa)) = numera6620942414471956472nteger(Xa) ).

% max_0_1(3)
tff(fact_4524_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),aa_num_Extended_enat(numera1916890842035813515d_enat,Xa)) = aa_num_Extended_enat(numera1916890842035813515d_enat,Xa) ).

% max_0_1(3)
tff(fact_4525_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,Xa)) = aa_num_real(numeral_numeral_real,Xa) ).

% max_0_1(3)
tff(fact_4526_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(3)
tff(fact_4527_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_int_int(ord_max_int(zero_zero_int),aa_num_int(numeral_numeral_int,Xa)) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(3)
tff(fact_4528_max__0__1_I3_J,axiom,
    ! [Xa: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,Xa)) = aa_num_rat(numeral_numeral_rat,Xa) ).

% max_0_1(3)
tff(fact_4529_max__0__1_I4_J,axiom,
    ! [Xa: num] : ord_max_Code_integer(numera6620942414471956472nteger(Xa),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(Xa) ).

% max_0_1(4)
tff(fact_4530_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Xa)),zero_z5237406670263579293d_enat) = aa_num_Extended_enat(numera1916890842035813515d_enat,Xa) ).

% max_0_1(4)
tff(fact_4531_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xa)),zero_zero_real) = aa_num_real(numeral_numeral_real,Xa) ).

% max_0_1(4)
tff(fact_4532_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xa)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(4)
tff(fact_4533_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,Xa)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(4)
tff(fact_4534_max__0__1_I4_J,axiom,
    ! [Xa: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xa)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,Xa) ).

% max_0_1(4)
tff(fact_4535_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_4536_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_4537_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_4538_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_4539_max__0__1_I2_J,axiom,
    aa_int_int(ord_max_int(one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_4540_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_4541_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_4542_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_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_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_4545_max__0__1_I1_J,axiom,
    aa_int_int(ord_max_int(zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_4546_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_4547_max__0__1_I5_J,axiom,
    ! [Xa: num] : ord_max_Code_integer(one_one_Code_integer,numera6620942414471956472nteger(Xa)) = numera6620942414471956472nteger(Xa) ).

% max_0_1(5)
tff(fact_4548_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),aa_num_Extended_enat(numera1916890842035813515d_enat,Xa)) = aa_num_Extended_enat(numera1916890842035813515d_enat,Xa) ).

% max_0_1(5)
tff(fact_4549_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,Xa)) = aa_num_real(numeral_numeral_real,Xa) ).

% max_0_1(5)
tff(fact_4550_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xa)) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(5)
tff(fact_4551_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_int_int(ord_max_int(one_one_int),aa_num_int(numeral_numeral_int,Xa)) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(5)
tff(fact_4552_max__0__1_I5_J,axiom,
    ! [Xa: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xa)) = aa_num_rat(numeral_numeral_rat,Xa) ).

% max_0_1(5)
tff(fact_4553_max__0__1_I6_J,axiom,
    ! [Xa: num] : ord_max_Code_integer(numera6620942414471956472nteger(Xa),one_one_Code_integer) = numera6620942414471956472nteger(Xa) ).

% max_0_1(6)
tff(fact_4554_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_num_Extended_enat(numera1916890842035813515d_enat,Xa)),one_on7984719198319812577d_enat) = aa_num_Extended_enat(numera1916890842035813515d_enat,Xa) ).

% max_0_1(6)
tff(fact_4555_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xa)),one_one_real) = aa_num_real(numeral_numeral_real,Xa) ).

% max_0_1(6)
tff(fact_4556_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xa) ).

% max_0_1(6)
tff(fact_4557_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,Xa)),one_one_int) = aa_num_int(numeral_numeral_int,Xa) ).

% max_0_1(6)
tff(fact_4558_max__0__1_I6_J,axiom,
    ! [Xa: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xa)),one_one_rat) = aa_num_rat(numeral_numeral_rat,Xa) ).

% max_0_1(6)
tff(fact_4559_Collect__disj__eq,axiom,
    ! [P: fun_real_o,Q: fun_real_o] : collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_cs(P),Q)) = sup_sup_set_real(collect_real(P),collect_real(Q)) ).

% Collect_disj_eq
tff(fact_4560_Collect__disj__eq,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ct(P),Q)) = sup_sup_set_list_nat(collect_list_nat(P),collect_list_nat(Q)) ).

% Collect_disj_eq
tff(fact_4561_Collect__disj__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_cu(P),Q)) = sup_sup_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ).

% Collect_disj_eq
tff(fact_4562_Collect__disj__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cv(P),Q)) = sup_sup_set_int(collect_int(P),collect_int(Q)) ).

% Collect_disj_eq
tff(fact_4563_Collect__disj__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cw(P),Q)) = sup_sup_set_nat(collect_nat(P),collect_nat(Q)) ).

% Collect_disj_eq
tff(fact_4564_Un__def,axiom,
    ! [A2: set_complex,B2: set_complex] : sup_sup_set_complex(A2,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_cx(A2),B2)) ).

% Un_def
tff(fact_4565_Un__def,axiom,
    ! [A2: set_real,B2: set_real] : sup_sup_set_real(A2,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_cy(A2),B2)) ).

% Un_def
tff(fact_4566_Un__def,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : sup_sup_set_list_nat(A2,B2) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_cz(A2),B2)) ).

% Un_def
tff(fact_4567_Un__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : sup_sup_set_set_nat(A2,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_da(A2),B2)) ).

% Un_def
tff(fact_4568_Un__def,axiom,
    ! [A2: set_int,B2: set_int] : sup_sup_set_int(A2,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_db(A2),B2)) ).

% Un_def
tff(fact_4569_Un__def,axiom,
    ! [A2: set_nat,B2: set_nat] : sup_sup_set_nat(A2,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_dc(A2),B2)) ).

% Un_def
tff(fact_4570_UnE,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),sup_sup_set_complex(A2,B2))
     => ( ~ aa_set_complex_o(member_complex(C),A2)
       => aa_set_complex_o(member_complex(C),B2) ) ) ).

% UnE
tff(fact_4571_UnE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2))
     => ( ~ aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B2) ) ) ).

% UnE
tff(fact_4572_UnE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2))
     => ( ~ aa_set_set_nat_o(member_set_nat(C),A2)
       => aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% UnE
tff(fact_4573_UnE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2))
     => ( ~ aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B2) ) ) ).

% UnE
tff(fact_4574_UnE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),sup_sup_set_nat(A2,B2))
     => ( ~ aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B2) ) ) ).

% UnE
tff(fact_4575_UnI1,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),A2)
     => aa_set_complex_o(member_complex(C),sup_sup_set_complex(A2,B2)) ) ).

% UnI1
tff(fact_4576_UnI1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),A2)
     => aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2)) ) ).

% UnI1
tff(fact_4577_UnI1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),A2)
     => aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2)) ) ).

% UnI1
tff(fact_4578_UnI1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2)) ) ).

% UnI1
tff(fact_4579_UnI1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => aa_set_nat_o(member_nat(C),sup_sup_set_nat(A2,B2)) ) ).

% UnI1
tff(fact_4580_UnI2,axiom,
    ! [C: complex,B2: set_complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(C),B2)
     => aa_set_complex_o(member_complex(C),sup_sup_set_complex(A2,B2)) ) ).

% UnI2
tff(fact_4581_UnI2,axiom,
    ! [C: real,B2: set_real,A2: set_real] :
      ( aa_set_real_o(member_real(C),B2)
     => aa_set_real_o(member_real(C),sup_sup_set_real(A2,B2)) ) ).

% UnI2
tff(fact_4582_UnI2,axiom,
    ! [C: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),B2)
     => aa_set_set_nat_o(member_set_nat(C),sup_sup_set_set_nat(A2,B2)) ) ).

% UnI2
tff(fact_4583_UnI2,axiom,
    ! [C: int,B2: set_int,A2: set_int] :
      ( aa_set_int_o(member_int(C),B2)
     => aa_set_int_o(member_int(C),sup_sup_set_int(A2,B2)) ) ).

% UnI2
tff(fact_4584_UnI2,axiom,
    ! [C: nat,B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C),B2)
     => aa_set_nat_o(member_nat(C),sup_sup_set_nat(A2,B2)) ) ).

% UnI2
tff(fact_4585_bex__Un,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),sup_sup_set_nat(A2,B2))
          & aa_nat_o(P,X4) )
    <=> ( ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & aa_nat_o(P,X4) )
        | ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B2)
            & aa_nat_o(P,X4) ) ) ) ).

% bex_Un
tff(fact_4586_ball__Un,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),sup_sup_set_nat(A2,B2))
         => aa_nat_o(P,X4) )
    <=> ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
           => aa_nat_o(P,X4) )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B2)
           => aa_nat_o(P,X4) ) ) ) ).

% ball_Un
tff(fact_4587_Un__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : sup_sup_set_nat(sup_sup_set_nat(A2,B2),C5) = sup_sup_set_nat(A2,sup_sup_set_nat(B2,C5)) ).

% Un_assoc
tff(fact_4588_Un__absorb,axiom,
    ! [A2: set_nat] : sup_sup_set_nat(A2,A2) = A2 ).

% Un_absorb
tff(fact_4589_Un__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : sup_sup_set_nat(A2,B2) = sup_sup_set_nat(B2,A2) ).

% Un_commute
tff(fact_4590_Un__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : sup_sup_set_nat(A2,sup_sup_set_nat(A2,B2)) = sup_sup_set_nat(A2,B2) ).

% Un_left_absorb
tff(fact_4591_Un__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : sup_sup_set_nat(A2,sup_sup_set_nat(B2,C5)) = sup_sup_set_nat(B2,sup_sup_set_nat(A2,C5)) ).

% Un_left_commute
tff(fact_4592_max__add__distrib__right,axiom,
    ! [Xa: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Z)) ).

% max_add_distrib_right
tff(fact_4593_max__add__distrib__right,axiom,
    ! [Xa: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xa),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xa),Y)),aa_rat_rat(plus_plus_rat(Xa),Z)) ).

% max_add_distrib_right
tff(fact_4594_max__add__distrib__right,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Z)) ).

% max_add_distrib_right
tff(fact_4595_max__add__distrib__right,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(plus_plus_int(Xa),aa_int_int(ord_max_int(Y),Z)) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(Xa),Y)),aa_int_int(plus_plus_int(Xa),Z)) ).

% max_add_distrib_right
tff(fact_4596_max__add__distrib__right,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(Xa,ord_max_Code_integer(Y,Z)) = ord_max_Code_integer(plus_p5714425477246183910nteger(Xa,Y),plus_p5714425477246183910nteger(Xa,Z)) ).

% max_add_distrib_right
tff(fact_4597_max__add__distrib__left,axiom,
    ! [Xa: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xa),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_4598_max__add__distrib__left,axiom,
    ! [Xa: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xa),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xa),Z)),aa_rat_rat(plus_plus_rat(Y),Z)) ).

% max_add_distrib_left
tff(fact_4599_max__add__distrib__left,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_4600_max__add__distrib__left,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(plus_plus_int(aa_int_int(ord_max_int(Xa),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(Xa),Z)),aa_int_int(plus_plus_int(Y),Z)) ).

% max_add_distrib_left
tff(fact_4601_max__add__distrib__left,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(ord_max_Code_integer(Xa,Y),Z) = ord_max_Code_integer(plus_p5714425477246183910nteger(Xa,Z),plus_p5714425477246183910nteger(Y,Z)) ).

% max_add_distrib_left
tff(fact_4602_max__diff__distrib__left,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] : minus_8373710615458151222nteger(ord_max_Code_integer(Xa,Y),Z) = ord_max_Code_integer(minus_8373710615458151222nteger(Xa,Z),minus_8373710615458151222nteger(Y,Z)) ).

% max_diff_distrib_left
tff(fact_4603_max__diff__distrib__left,axiom,
    ! [Xa: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xa),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_4604_max__diff__distrib__left,axiom,
    ! [Xa: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(Xa),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(Xa),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).

% max_diff_distrib_left
tff(fact_4605_max__diff__distrib__left,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(minus_minus_int(aa_int_int(ord_max_int(Xa),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(minus_minus_int(Xa),Z)),aa_int_int(minus_minus_int(Y),Z)) ).

% max_diff_distrib_left
tff(fact_4606_nat__add__max__right,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Q2)) ).

% nat_add_max_right
tff(fact_4607_nat__add__max__left,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q2)) ).

% nat_add_max_left
tff(fact_4608_nat__mult__max__left,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_max_left
tff(fact_4609_nat__mult__max__right,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Q2)) ).

% nat_mult_max_right
tff(fact_4610_Un__empty__right,axiom,
    ! [A2: set_real] : sup_sup_set_real(A2,bot_bot_set_real) = A2 ).

% Un_empty_right
tff(fact_4611_Un__empty__right,axiom,
    ! [A2: set_o] : sup_sup_set_o(A2,bot_bot_set_o) = A2 ).

% Un_empty_right
tff(fact_4612_Un__empty__right,axiom,
    ! [A2: set_nat] : sup_sup_set_nat(A2,bot_bot_set_nat) = A2 ).

% Un_empty_right
tff(fact_4613_Un__empty__right,axiom,
    ! [A2: set_int] : sup_sup_set_int(A2,bot_bot_set_int) = A2 ).

% Un_empty_right
tff(fact_4614_Un__empty__left,axiom,
    ! [B2: set_real] : sup_sup_set_real(bot_bot_set_real,B2) = B2 ).

% Un_empty_left
tff(fact_4615_Un__empty__left,axiom,
    ! [B2: set_o] : sup_sup_set_o(bot_bot_set_o,B2) = B2 ).

% Un_empty_left
tff(fact_4616_Un__empty__left,axiom,
    ! [B2: set_nat] : sup_sup_set_nat(bot_bot_set_nat,B2) = B2 ).

% Un_empty_left
tff(fact_4617_Un__empty__left,axiom,
    ! [B2: set_int] : sup_sup_set_int(bot_bot_set_int,B2) = B2 ).

% Un_empty_left
tff(fact_4618_subset__Un__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ( sup_sup_set_nat(A2,B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_4619_subset__UnE,axiom,
    ! [C5: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C5),sup_sup_set_nat(A2,B2))
     => ~ ! [A6: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(A6),A2)
           => ! [B8: set_nat] :
                ( aa_set_nat_o(ord_less_eq_set_nat(B8),B2)
               => ( C5 != sup_sup_set_nat(A6,B8) ) ) ) ) ).

% subset_UnE
tff(fact_4620_Un__absorb2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( sup_sup_set_nat(A2,B2) = A2 ) ) ).

% Un_absorb2
tff(fact_4621_Un__absorb1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( sup_sup_set_nat(A2,B2) = B2 ) ) ).

% Un_absorb1
tff(fact_4622_Un__upper2,axiom,
    ! [B2: set_nat,A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(B2),sup_sup_set_nat(A2,B2)) ).

% Un_upper2
tff(fact_4623_Un__upper1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A2),sup_sup_set_nat(A2,B2)) ).

% Un_upper1
tff(fact_4624_Un__least,axiom,
    ! [A2: set_nat,C5: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C5)
       => aa_set_nat_o(ord_less_eq_set_nat(sup_sup_set_nat(A2,B2)),C5) ) ) ).

% Un_least
tff(fact_4625_Un__mono,axiom,
    ! [A2: set_nat,C5: set_nat,B2: set_nat,D4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),D4)
       => aa_set_nat_o(ord_less_eq_set_nat(sup_sup_set_nat(A2,B2)),sup_sup_set_nat(C5,D4)) ) ) ).

% Un_mono
tff(fact_4626_Un__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : minus_minus_set_nat(sup_sup_set_nat(A2,B2),C5) = sup_sup_set_nat(minus_minus_set_nat(A2,C5),minus_minus_set_nat(B2,C5)) ).

% Un_Diff
tff(fact_4627_max__def__raw,axiom,
    ! [X: extended_enat,Xa2: extended_enat] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Xa2) = $ite(aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,X),Xa2),Xa2,X) ).

% max_def_raw
tff(fact_4628_max__def__raw,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ord_max_Code_integer(X,Xa2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa2),Xa2,X) ).

% max_def_raw
tff(fact_4629_max__def__raw,axiom,
    ! [X: set_nat,Xa2: set_nat] :
      ord_max_set_nat(X,Xa2) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X),Xa2),Xa2,X) ).

% max_def_raw
tff(fact_4630_max__def__raw,axiom,
    ! [X: rat,Xa2: rat] :
      aa_rat_rat(ord_max_rat(X),Xa2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa2),Xa2,X) ).

% max_def_raw
tff(fact_4631_max__def__raw,axiom,
    ! [X: num,Xa2: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X),Xa2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa2),Xa2,X) ).

% max_def_raw
tff(fact_4632_max__def__raw,axiom,
    ! [X: nat,Xa2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa2),Xa2,X) ).

% max_def_raw
tff(fact_4633_max__def__raw,axiom,
    ! [X: int,Xa2: int] :
      aa_int_int(ord_max_int(X),Xa2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2),Xa2,X) ).

% max_def_raw
tff(fact_4634_insert__def,axiom,
    ! [A: $o,B2: set_o] : insert_o((A),B2) = sup_sup_set_o(collect_o(aTP_Lamm_ak((A))),B2) ).

% insert_def
tff(fact_4635_insert__def,axiom,
    ! [A: real,B2: set_real] : insert_real(A,B2) = sup_sup_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_aj,A)),B2) ).

% insert_def
tff(fact_4636_insert__def,axiom,
    ! [A: list_nat,B2: set_list_nat] : insert_list_nat(A,B2) = sup_sup_set_list_nat(collect_list_nat(aTP_Lamm_ah(A)),B2) ).

% insert_def
tff(fact_4637_insert__def,axiom,
    ! [A: set_nat,B2: set_set_nat] : insert_set_nat(A,B2) = sup_sup_set_set_nat(collect_set_nat(aTP_Lamm_ai(A)),B2) ).

% insert_def
tff(fact_4638_insert__def,axiom,
    ! [A: int,B2: set_int] : insert_int(A,B2) = sup_sup_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_am,A)),B2) ).

% insert_def
tff(fact_4639_insert__def,axiom,
    ! [A: nat,B2: set_nat] : insert_nat(A,B2) = sup_sup_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_al,A)),B2) ).

% insert_def
tff(fact_4640_nat__minus__add__max,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Ma) ).

% nat_minus_add_max
tff(fact_4641_singleton__Un__iff,axiom,
    ! [Xa: real,A2: set_real,B2: set_real] :
      ( ( insert_real(Xa,bot_bot_set_real) = sup_sup_set_real(A2,B2) )
    <=> ( ( ( A2 = bot_bot_set_real )
          & ( B2 = insert_real(Xa,bot_bot_set_real) ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B2 = insert_real(Xa,bot_bot_set_real) ) ) ) ) ).

% singleton_Un_iff
tff(fact_4642_singleton__Un__iff,axiom,
    ! [Xa: $o,A2: set_o,B2: set_o] :
      ( ( insert_o((Xa),bot_bot_set_o) = sup_sup_set_o(A2,B2) )
    <=> ( ( ( A2 = bot_bot_set_o )
          & ( B2 = insert_o((Xa),bot_bot_set_o) ) )
        | ( ( A2 = insert_o((Xa),bot_bot_set_o) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A2 = insert_o((Xa),bot_bot_set_o) )
          & ( B2 = insert_o((Xa),bot_bot_set_o) ) ) ) ) ).

% singleton_Un_iff
tff(fact_4643_singleton__Un__iff,axiom,
    ! [Xa: nat,A2: set_nat,B2: set_nat] :
      ( ( insert_nat(Xa,bot_bot_set_nat) = sup_sup_set_nat(A2,B2) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B2 = insert_nat(Xa,bot_bot_set_nat) ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B2 = insert_nat(Xa,bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_4644_singleton__Un__iff,axiom,
    ! [Xa: int,A2: set_int,B2: set_int] :
      ( ( insert_int(Xa,bot_bot_set_int) = sup_sup_set_int(A2,B2) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B2 = insert_int(Xa,bot_bot_set_int) ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B2 = insert_int(Xa,bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_4645_Un__singleton__iff,axiom,
    ! [A2: set_real,B2: set_real,Xa: real] :
      ( ( sup_sup_set_real(A2,B2) = insert_real(Xa,bot_bot_set_real) )
    <=> ( ( ( A2 = bot_bot_set_real )
          & ( B2 = insert_real(Xa,bot_bot_set_real) ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A2 = insert_real(Xa,bot_bot_set_real) )
          & ( B2 = insert_real(Xa,bot_bot_set_real) ) ) ) ) ).

% Un_singleton_iff
tff(fact_4646_Un__singleton__iff,axiom,
    ! [A2: set_o,B2: set_o,Xa: $o] :
      ( ( sup_sup_set_o(A2,B2) = insert_o((Xa),bot_bot_set_o) )
    <=> ( ( ( A2 = bot_bot_set_o )
          & ( B2 = insert_o((Xa),bot_bot_set_o) ) )
        | ( ( A2 = insert_o((Xa),bot_bot_set_o) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A2 = insert_o((Xa),bot_bot_set_o) )
          & ( B2 = insert_o((Xa),bot_bot_set_o) ) ) ) ) ).

% Un_singleton_iff
tff(fact_4647_Un__singleton__iff,axiom,
    ! [A2: set_nat,B2: set_nat,Xa: nat] :
      ( ( sup_sup_set_nat(A2,B2) = insert_nat(Xa,bot_bot_set_nat) )
    <=> ( ( ( A2 = bot_bot_set_nat )
          & ( B2 = insert_nat(Xa,bot_bot_set_nat) ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2 = insert_nat(Xa,bot_bot_set_nat) )
          & ( B2 = insert_nat(Xa,bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_4648_Un__singleton__iff,axiom,
    ! [A2: set_int,B2: set_int,Xa: int] :
      ( ( sup_sup_set_int(A2,B2) = insert_int(Xa,bot_bot_set_int) )
    <=> ( ( ( A2 = bot_bot_set_int )
          & ( B2 = insert_int(Xa,bot_bot_set_int) ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A2 = insert_int(Xa,bot_bot_set_int) )
          & ( B2 = insert_int(Xa,bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_4649_insert__is__Un,axiom,
    ! [A: real,A2: set_real] : insert_real(A,A2) = sup_sup_set_real(insert_real(A,bot_bot_set_real),A2) ).

% insert_is_Un
tff(fact_4650_insert__is__Un,axiom,
    ! [A: $o,A2: set_o] : insert_o((A),A2) = sup_sup_set_o(insert_o((A),bot_bot_set_o),A2) ).

% insert_is_Un
tff(fact_4651_insert__is__Un,axiom,
    ! [A: nat,A2: set_nat] : insert_nat(A,A2) = sup_sup_set_nat(insert_nat(A,bot_bot_set_nat),A2) ).

% insert_is_Un
tff(fact_4652_insert__is__Un,axiom,
    ! [A: int,A2: set_int] : insert_int(A,A2) = sup_sup_set_int(insert_int(A,bot_bot_set_int),A2) ).

% insert_is_Un
tff(fact_4653_Diff__partition,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( sup_sup_set_nat(A2,minus_minus_set_nat(B2,A2)) = B2 ) ) ).

% Diff_partition
tff(fact_4654_Diff__subset__conv,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,B2)),C5)
    <=> aa_set_nat_o(ord_less_eq_set_nat(A2),sup_sup_set_nat(B2,C5)) ) ).

% Diff_subset_conv
tff(fact_4655_simp__from__to,axiom,
    ! [Ia: int,J: int] :
      set_or1266510415728281911st_int(Ia,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia),bot_bot_set_int,insert_int(Ia,set_or1266510415728281911st_int(aa_int_int(plus_plus_int(Ia),one_one_int),J))) ).

% simp_from_to
tff(fact_4656_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Xa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_4657_vebt__delete_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = vEBT_Leaf($false,(B4)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf((A4),$false) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B4)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = vEBT_Leaf((A4),(B4)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary2),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( Xa = 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),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( Xa = 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),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = 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(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi2,xn,Mi2),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                              $let(
                                                newlist: list_VEBT_VEBT,
                                                newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                $ite(
                                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary2,h),
                                                    vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                          $ite(
                                                            xn = Ma2,
                                                            $let(
                                                              maxs: option_nat,
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none_nat,minn,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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                            Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,sn) ),
                                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                        $ite(xn = Ma2,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,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,Summary2) ) ) ),
                                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,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)),TreeList2,Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_4658_vebt__insert_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B4)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Xaa,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,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)),TreeList2,Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4659_insert_H__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xa: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(vEBT_VEBT_insert(Ta,Xa)) = inf_inf_set_nat(sup_sup_set_nat(vEBT_set_vebt(Ta),insert_nat(Xa,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))),Nb)),one_one_nat))) ) ) ).

% insert'_correct
tff(fact_4660_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = vEBT_vebt_insert(vEBT_Leaf((A4),(B4)),Xaa) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg,TreeList2,Summary2) )
               => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)),Xaa),vEBT_Node(Info2,Deg,TreeList2,Summary2),vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList2,Summary2),Xaa)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,Deg,TreeList2,Summary2),Xaa)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_4661_vebt__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B4),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi2,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4662_vebt__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,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)),TreeList2,Summary2),Xaa))
                       => $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4663_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B4),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4664_Int__iff,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),inf_inf_set_complex(A2,B2))
    <=> ( aa_set_complex_o(member_complex(C),A2)
        & aa_set_complex_o(member_complex(C),B2) ) ) ).

% Int_iff
tff(fact_4665_Int__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2))
    <=> ( aa_set_real_o(member_real(C),A2)
        & aa_set_real_o(member_real(C),B2) ) ) ).

% Int_iff
tff(fact_4666_Int__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2))
    <=> ( aa_set_set_nat_o(member_set_nat(C),A2)
        & aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% Int_iff
tff(fact_4667_Int__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2))
    <=> ( aa_set_int_o(member_int(C),A2)
        & aa_set_int_o(member_int(C),B2) ) ) ).

% Int_iff
tff(fact_4668_Int__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),inf_inf_set_nat(A2,B2))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        & aa_set_nat_o(member_nat(C),B2) ) ) ).

% Int_iff
tff(fact_4669_IntI,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),A2)
     => ( aa_set_complex_o(member_complex(C),B2)
       => aa_set_complex_o(member_complex(C),inf_inf_set_complex(A2,B2)) ) ) ).

% IntI
tff(fact_4670_IntI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),A2)
     => ( aa_set_real_o(member_real(C),B2)
       => aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2)) ) ) ).

% IntI
tff(fact_4671_IntI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),A2)
     => ( aa_set_set_nat_o(member_set_nat(C),B2)
       => aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2)) ) ) ).

% IntI
tff(fact_4672_IntI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),A2)
     => ( aa_set_int_o(member_int(C),B2)
       => aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2)) ) ) ).

% IntI
tff(fact_4673_IntI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => ( aa_set_nat_o(member_nat(C),B2)
       => aa_set_nat_o(member_nat(C),inf_inf_set_nat(A2,B2)) ) ) ).

% IntI
tff(fact_4674_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q2),zero_z5237406670263579293d_enat) = Q2 ).

% max_enat_simps(2)
tff(fact_4675_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_4676_Int__subset__iff,axiom,
    ! [C5: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C5),inf_inf_set_nat(A2,B2))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(C5),A2)
        & aa_set_nat_o(ord_less_eq_set_nat(C5),B2) ) ) ).

% Int_subset_iff
tff(fact_4677_Int__insert__right__if1,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( aa_set_o_o(member_o((A)),A2)
     => ( inf_inf_set_o(A2,insert_o((A),B2)) = insert_o((A),inf_inf_set_o(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4678_Int__insert__right__if1,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(A),A2)
     => ( inf_inf_set_complex(A2,insert_complex(A,B2)) = insert_complex(A,inf_inf_set_complex(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4679_Int__insert__right__if1,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(A),A2)
     => ( inf_inf_set_real(A2,insert_real(A,B2)) = insert_real(A,inf_inf_set_real(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4680_Int__insert__right__if1,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A2)
     => ( inf_inf_set_set_nat(A2,insert_set_nat(A,B2)) = insert_set_nat(A,inf_inf_set_set_nat(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4681_Int__insert__right__if1,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(A),A2)
     => ( inf_inf_set_int(A2,insert_int(A,B2)) = insert_int(A,inf_inf_set_int(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4682_Int__insert__right__if1,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(A),A2)
     => ( inf_inf_set_nat(A2,insert_nat(A,B2)) = insert_nat(A,inf_inf_set_nat(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4683_Int__insert__right__if0,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A2)
     => ( inf_inf_set_o(A2,insert_o((A),B2)) = inf_inf_set_o(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_4684_Int__insert__right__if0,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(A),A2)
     => ( inf_inf_set_complex(A2,insert_complex(A,B2)) = inf_inf_set_complex(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_4685_Int__insert__right__if0,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(A),A2)
     => ( inf_inf_set_real(A2,insert_real(A,B2)) = inf_inf_set_real(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_4686_Int__insert__right__if0,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A2)
     => ( inf_inf_set_set_nat(A2,insert_set_nat(A,B2)) = inf_inf_set_set_nat(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_4687_Int__insert__right__if0,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),A2)
     => ( inf_inf_set_int(A2,insert_int(A,B2)) = inf_inf_set_int(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_4688_Int__insert__right__if0,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A2)
     => ( inf_inf_set_nat(A2,insert_nat(A,B2)) = inf_inf_set_nat(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_4689_insert__inter__insert,axiom,
    ! [A: int,A2: set_int,B2: set_int] : inf_inf_set_int(insert_int(A,A2),insert_int(A,B2)) = insert_int(A,inf_inf_set_int(A2,B2)) ).

% insert_inter_insert
tff(fact_4690_insert__inter__insert,axiom,
    ! [A: real,A2: set_real,B2: set_real] : inf_inf_set_real(insert_real(A,A2),insert_real(A,B2)) = insert_real(A,inf_inf_set_real(A2,B2)) ).

% insert_inter_insert
tff(fact_4691_insert__inter__insert,axiom,
    ! [A: $o,A2: set_o,B2: set_o] : inf_inf_set_o(insert_o((A),A2),insert_o((A),B2)) = insert_o((A),inf_inf_set_o(A2,B2)) ).

% insert_inter_insert
tff(fact_4692_insert__inter__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] : inf_inf_set_nat(insert_nat(A,A2),insert_nat(A,B2)) = insert_nat(A,inf_inf_set_nat(A2,B2)) ).

% insert_inter_insert
tff(fact_4693_Int__insert__left__if1,axiom,
    ! [A: $o,C5: set_o,B2: set_o] :
      ( aa_set_o_o(member_o((A)),C5)
     => ( inf_inf_set_o(insert_o((A),B2),C5) = insert_o((A),inf_inf_set_o(B2,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_4694_Int__insert__left__if1,axiom,
    ! [A: complex,C5: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(A),C5)
     => ( inf_inf_set_complex(insert_complex(A,B2),C5) = insert_complex(A,inf_inf_set_complex(B2,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_4695_Int__insert__left__if1,axiom,
    ! [A: real,C5: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(A),C5)
     => ( inf_inf_set_real(insert_real(A,B2),C5) = insert_real(A,inf_inf_set_real(B2,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_4696_Int__insert__left__if1,axiom,
    ! [A: set_nat,C5: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),C5)
     => ( inf_inf_set_set_nat(insert_set_nat(A,B2),C5) = insert_set_nat(A,inf_inf_set_set_nat(B2,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_4697_Int__insert__left__if1,axiom,
    ! [A: int,C5: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(A),C5)
     => ( inf_inf_set_int(insert_int(A,B2),C5) = insert_int(A,inf_inf_set_int(B2,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_4698_Int__insert__left__if1,axiom,
    ! [A: nat,C5: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(A),C5)
     => ( inf_inf_set_nat(insert_nat(A,B2),C5) = insert_nat(A,inf_inf_set_nat(B2,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_4699_Int__insert__left__if0,axiom,
    ! [A: $o,C5: set_o,B2: set_o] :
      ( ~ aa_set_o_o(member_o((A)),C5)
     => ( inf_inf_set_o(insert_o((A),B2),C5) = inf_inf_set_o(B2,C5) ) ) ).

% Int_insert_left_if0
tff(fact_4700_Int__insert__left__if0,axiom,
    ! [A: complex,C5: set_complex,B2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(A),C5)
     => ( inf_inf_set_complex(insert_complex(A,B2),C5) = inf_inf_set_complex(B2,C5) ) ) ).

% Int_insert_left_if0
tff(fact_4701_Int__insert__left__if0,axiom,
    ! [A: real,C5: set_real,B2: set_real] :
      ( ~ aa_set_real_o(member_real(A),C5)
     => ( inf_inf_set_real(insert_real(A,B2),C5) = inf_inf_set_real(B2,C5) ) ) ).

% Int_insert_left_if0
tff(fact_4702_Int__insert__left__if0,axiom,
    ! [A: set_nat,C5: set_set_nat,B2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),C5)
     => ( inf_inf_set_set_nat(insert_set_nat(A,B2),C5) = inf_inf_set_set_nat(B2,C5) ) ) ).

% Int_insert_left_if0
tff(fact_4703_Int__insert__left__if0,axiom,
    ! [A: int,C5: set_int,B2: set_int] :
      ( ~ aa_set_int_o(member_int(A),C5)
     => ( inf_inf_set_int(insert_int(A,B2),C5) = inf_inf_set_int(B2,C5) ) ) ).

% Int_insert_left_if0
tff(fact_4704_Int__insert__left__if0,axiom,
    ! [A: nat,C5: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),C5)
     => ( inf_inf_set_nat(insert_nat(A,B2),C5) = inf_inf_set_nat(B2,C5) ) ) ).

% Int_insert_left_if0
tff(fact_4705_Un__Int__eq_I1_J,axiom,
    ! [S2: set_nat,T2: set_nat] : inf_inf_set_nat(sup_sup_set_nat(S2,T2),S2) = S2 ).

% Un_Int_eq(1)
tff(fact_4706_Un__Int__eq_I2_J,axiom,
    ! [S2: set_nat,T2: set_nat] : inf_inf_set_nat(sup_sup_set_nat(S2,T2),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_4707_Un__Int__eq_I3_J,axiom,
    ! [S2: set_nat,T2: set_nat] : inf_inf_set_nat(S2,sup_sup_set_nat(S2,T2)) = S2 ).

% Un_Int_eq(3)
tff(fact_4708_Un__Int__eq_I4_J,axiom,
    ! [T2: set_nat,S2: set_nat] : inf_inf_set_nat(T2,sup_sup_set_nat(S2,T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_4709_Int__Un__eq_I1_J,axiom,
    ! [S2: set_nat,T2: set_nat] : sup_sup_set_nat(inf_inf_set_nat(S2,T2),S2) = S2 ).

% Int_Un_eq(1)
tff(fact_4710_Int__Un__eq_I2_J,axiom,
    ! [S2: set_nat,T2: set_nat] : sup_sup_set_nat(inf_inf_set_nat(S2,T2),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_4711_Int__Un__eq_I3_J,axiom,
    ! [S2: set_nat,T2: set_nat] : sup_sup_set_nat(S2,inf_inf_set_nat(S2,T2)) = S2 ).

% Int_Un_eq(3)
tff(fact_4712_Int__Un__eq_I4_J,axiom,
    ! [T2: set_nat,S2: set_nat] : sup_sup_set_nat(T2,inf_inf_set_nat(S2,T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_4713_disjoint__insert_I2_J,axiom,
    ! [A2: set_complex,B: complex,B2: set_complex] :
      ( ( bot_bot_set_complex = inf_inf_set_complex(A2,insert_complex(B,B2)) )
    <=> ( ~ aa_set_complex_o(member_complex(B),A2)
        & ( bot_bot_set_complex = inf_inf_set_complex(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4714_disjoint__insert_I2_J,axiom,
    ! [A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,insert_set_nat(B,B2)) )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(B),A2)
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4715_disjoint__insert_I2_J,axiom,
    ! [A2: set_real,B: real,B2: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(A2,insert_real(B,B2)) )
    <=> ( ~ aa_set_real_o(member_real(B),A2)
        & ( bot_bot_set_real = inf_inf_set_real(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4716_disjoint__insert_I2_J,axiom,
    ! [A2: set_o,B: $o,B2: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(A2,insert_o((B),B2)) )
    <=> ( ~ aa_set_o_o(member_o((B)),A2)
        & ( bot_bot_set_o = inf_inf_set_o(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4717_disjoint__insert_I2_J,axiom,
    ! [A2: set_nat,B: nat,B2: set_nat] :
      ( ( bot_bot_set_nat = inf_inf_set_nat(A2,insert_nat(B,B2)) )
    <=> ( ~ aa_set_nat_o(member_nat(B),A2)
        & ( bot_bot_set_nat = inf_inf_set_nat(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4718_disjoint__insert_I2_J,axiom,
    ! [A2: set_int,B: int,B2: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(A2,insert_int(B,B2)) )
    <=> ( ~ aa_set_int_o(member_int(B),A2)
        & ( bot_bot_set_int = inf_inf_set_int(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4719_disjoint__insert_I1_J,axiom,
    ! [B2: set_complex,A: complex,A2: set_complex] :
      ( ( inf_inf_set_complex(B2,insert_complex(A,A2)) = bot_bot_set_complex )
    <=> ( ~ aa_set_complex_o(member_complex(A),B2)
        & ( inf_inf_set_complex(B2,A2) = bot_bot_set_complex ) ) ) ).

% disjoint_insert(1)
tff(fact_4720_disjoint__insert_I1_J,axiom,
    ! [B2: set_set_nat,A: set_nat,A2: set_set_nat] :
      ( ( inf_inf_set_set_nat(B2,insert_set_nat(A,A2)) = bot_bot_set_set_nat )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
        & ( inf_inf_set_set_nat(B2,A2) = bot_bot_set_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_4721_disjoint__insert_I1_J,axiom,
    ! [B2: set_real,A: real,A2: set_real] :
      ( ( inf_inf_set_real(B2,insert_real(A,A2)) = bot_bot_set_real )
    <=> ( ~ aa_set_real_o(member_real(A),B2)
        & ( inf_inf_set_real(B2,A2) = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
tff(fact_4722_disjoint__insert_I1_J,axiom,
    ! [B2: set_o,A: $o,A2: set_o] :
      ( ( inf_inf_set_o(B2,insert_o((A),A2)) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( inf_inf_set_o(B2,A2) = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
tff(fact_4723_disjoint__insert_I1_J,axiom,
    ! [B2: set_nat,A: nat,A2: set_nat] :
      ( ( inf_inf_set_nat(B2,insert_nat(A,A2)) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B2)
        & ( inf_inf_set_nat(B2,A2) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_4724_disjoint__insert_I1_J,axiom,
    ! [B2: set_int,A: int,A2: set_int] :
      ( ( inf_inf_set_int(B2,insert_int(A,A2)) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( inf_inf_set_int(B2,A2) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_4725_insert__disjoint_I2_J,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ( bot_bot_set_complex = inf_inf_set_complex(insert_complex(A,A2),B2) )
    <=> ( ~ aa_set_complex_o(member_complex(A),B2)
        & ( bot_bot_set_complex = inf_inf_set_complex(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4726_insert__disjoint_I2_J,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(insert_set_nat(A,A2),B2) )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4727_insert__disjoint_I2_J,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( bot_bot_set_real = inf_inf_set_real(insert_real(A,A2),B2) )
    <=> ( ~ aa_set_real_o(member_real(A),B2)
        & ( bot_bot_set_real = inf_inf_set_real(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4728_insert__disjoint_I2_J,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(insert_o((A),A2),B2) )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( bot_bot_set_o = inf_inf_set_o(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4729_insert__disjoint_I2_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat = inf_inf_set_nat(insert_nat(A,A2),B2) )
    <=> ( ~ aa_set_nat_o(member_nat(A),B2)
        & ( bot_bot_set_nat = inf_inf_set_nat(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4730_insert__disjoint_I2_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(insert_int(A,A2),B2) )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( bot_bot_set_int = inf_inf_set_int(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4731_insert__disjoint_I1_J,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ( inf_inf_set_complex(insert_complex(A,A2),B2) = bot_bot_set_complex )
    <=> ( ~ aa_set_complex_o(member_complex(A),B2)
        & ( inf_inf_set_complex(A2,B2) = bot_bot_set_complex ) ) ) ).

% insert_disjoint(1)
tff(fact_4732_insert__disjoint_I1_J,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( inf_inf_set_set_nat(insert_set_nat(A,A2),B2) = bot_bot_set_set_nat )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B2)
        & ( inf_inf_set_set_nat(A2,B2) = bot_bot_set_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_4733_insert__disjoint_I1_J,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(insert_real(A,A2),B2) = bot_bot_set_real )
    <=> ( ~ aa_set_real_o(member_real(A),B2)
        & ( inf_inf_set_real(A2,B2) = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
tff(fact_4734_insert__disjoint_I1_J,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(insert_o((A),A2),B2) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B2)
        & ( inf_inf_set_o(A2,B2) = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
tff(fact_4735_insert__disjoint_I1_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat(insert_nat(A,A2),B2) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B2)
        & ( inf_inf_set_nat(A2,B2) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_4736_insert__disjoint_I1_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(insert_int(A,A2),B2) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B2)
        & ( inf_inf_set_int(A2,B2) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_4737_Diff__disjoint,axiom,
    ! [A2: set_real,B2: set_real] : inf_inf_set_real(A2,minus_minus_set_real(B2,A2)) = bot_bot_set_real ).

% Diff_disjoint
tff(fact_4738_Diff__disjoint,axiom,
    ! [A2: set_o,B2: set_o] : inf_inf_set_o(A2,minus_minus_set_o(B2,A2)) = bot_bot_set_o ).

% Diff_disjoint
tff(fact_4739_Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] : inf_inf_set_int(A2,minus_minus_set_int(B2,A2)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_4740_Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : inf_inf_set_nat(A2,minus_minus_set_nat(B2,A2)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_4741_Int__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : inf_inf_set_nat(A2,inf_inf_set_nat(B2,C5)) = inf_inf_set_nat(B2,inf_inf_set_nat(A2,C5)) ).

% Int_left_commute
tff(fact_4742_Int__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : inf_inf_set_nat(A2,inf_inf_set_nat(A2,B2)) = inf_inf_set_nat(A2,B2) ).

% Int_left_absorb
tff(fact_4743_Int__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : inf_inf_set_nat(A2,B2) = inf_inf_set_nat(B2,A2) ).

% Int_commute
tff(fact_4744_Int__absorb,axiom,
    ! [A2: set_nat] : inf_inf_set_nat(A2,A2) = A2 ).

% Int_absorb
tff(fact_4745_Int__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : inf_inf_set_nat(inf_inf_set_nat(A2,B2),C5) = inf_inf_set_nat(A2,inf_inf_set_nat(B2,C5)) ).

% Int_assoc
tff(fact_4746_IntD2,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),inf_inf_set_complex(A2,B2))
     => aa_set_complex_o(member_complex(C),B2) ) ).

% IntD2
tff(fact_4747_IntD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2))
     => aa_set_real_o(member_real(C),B2) ) ).

% IntD2
tff(fact_4748_IntD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2))
     => aa_set_set_nat_o(member_set_nat(C),B2) ) ).

% IntD2
tff(fact_4749_IntD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2))
     => aa_set_int_o(member_int(C),B2) ) ).

% IntD2
tff(fact_4750_IntD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),inf_inf_set_nat(A2,B2))
     => aa_set_nat_o(member_nat(C),B2) ) ).

% IntD2
tff(fact_4751_IntD1,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),inf_inf_set_complex(A2,B2))
     => aa_set_complex_o(member_complex(C),A2) ) ).

% IntD1
tff(fact_4752_IntD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2))
     => aa_set_real_o(member_real(C),A2) ) ).

% IntD1
tff(fact_4753_IntD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2))
     => aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% IntD1
tff(fact_4754_IntD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2))
     => aa_set_int_o(member_int(C),A2) ) ).

% IntD1
tff(fact_4755_IntD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),inf_inf_set_nat(A2,B2))
     => aa_set_nat_o(member_nat(C),A2) ) ).

% IntD1
tff(fact_4756_IntE,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( aa_set_complex_o(member_complex(C),inf_inf_set_complex(A2,B2))
     => ~ ( aa_set_complex_o(member_complex(C),A2)
         => ~ aa_set_complex_o(member_complex(C),B2) ) ) ).

% IntE
tff(fact_4757_IntE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( aa_set_real_o(member_real(C),inf_inf_set_real(A2,B2))
     => ~ ( aa_set_real_o(member_real(C),A2)
         => ~ aa_set_real_o(member_real(C),B2) ) ) ).

% IntE
tff(fact_4758_IntE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),inf_inf_set_set_nat(A2,B2))
     => ~ ( aa_set_set_nat_o(member_set_nat(C),A2)
         => ~ aa_set_set_nat_o(member_set_nat(C),B2) ) ) ).

% IntE
tff(fact_4759_IntE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A2,B2))
     => ~ ( aa_set_int_o(member_int(C),A2)
         => ~ aa_set_int_o(member_int(C),B2) ) ) ).

% IntE
tff(fact_4760_IntE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(member_nat(C),inf_inf_set_nat(A2,B2))
     => ~ ( aa_set_nat_o(member_nat(C),A2)
         => ~ aa_set_nat_o(member_nat(C),B2) ) ) ).

% IntE
tff(fact_4761_Int__def,axiom,
    ! [A2: set_complex,B2: set_complex] : inf_inf_set_complex(A2,B2) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_dd(A2),B2)) ).

% Int_def
tff(fact_4762_Int__def,axiom,
    ! [A2: set_real,B2: set_real] : inf_inf_set_real(A2,B2) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_de(A2),B2)) ).

% Int_def
tff(fact_4763_Int__def,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : inf_inf_set_list_nat(A2,B2) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_df(A2),B2)) ).

% Int_def
tff(fact_4764_Int__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : inf_inf_set_set_nat(A2,B2) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_dg(A2),B2)) ).

% Int_def
tff(fact_4765_Int__def,axiom,
    ! [A2: set_int,B2: set_int] : inf_inf_set_int(A2,B2) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_dh(A2),B2)) ).

% Int_def
tff(fact_4766_Int__def,axiom,
    ! [A2: set_nat,B2: set_nat] : inf_inf_set_nat(A2,B2) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_di(A2),B2)) ).

% Int_def
tff(fact_4767_Int__Collect,axiom,
    ! [Xa: complex,A2: set_complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(Xa),inf_inf_set_complex(A2,collect_complex(P)))
    <=> ( aa_set_complex_o(member_complex(Xa),A2)
        & aa_complex_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_4768_Int__Collect,axiom,
    ! [Xa: real,A2: set_real,P: fun_real_o] :
      ( aa_set_real_o(member_real(Xa),inf_inf_set_real(A2,collect_real(P)))
    <=> ( aa_set_real_o(member_real(Xa),A2)
        & aa_real_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_4769_Int__Collect,axiom,
    ! [Xa: list_nat,A2: set_list_nat,P: fun_list_nat_o] :
      ( member_list_nat(Xa,inf_inf_set_list_nat(A2,collect_list_nat(P)))
    <=> ( member_list_nat(Xa,A2)
        & aa_list_nat_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_4770_Int__Collect,axiom,
    ! [Xa: set_nat,A2: set_set_nat,P: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(Xa),inf_inf_set_set_nat(A2,collect_set_nat(P)))
    <=> ( aa_set_set_nat_o(member_set_nat(Xa),A2)
        & aa_set_nat_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_4771_Int__Collect,axiom,
    ! [Xa: int,A2: set_int,P: fun_int_o] :
      ( aa_set_int_o(member_int(Xa),inf_inf_set_int(A2,collect_int(P)))
    <=> ( aa_set_int_o(member_int(Xa),A2)
        & aa_int_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_4772_Int__Collect,axiom,
    ! [Xa: nat,A2: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xa),inf_inf_set_nat(A2,collect_nat(P)))
    <=> ( aa_set_nat_o(member_nat(Xa),A2)
        & aa_nat_o(P,Xa) ) ) ).

% Int_Collect
tff(fact_4773_Collect__conj__eq,axiom,
    ! [P: fun_real_o,Q: fun_real_o] : collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_dj(P),Q)) = inf_inf_set_real(collect_real(P),collect_real(Q)) ).

% Collect_conj_eq
tff(fact_4774_Collect__conj__eq,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dk(P),Q)) = inf_inf_set_list_nat(collect_list_nat(P),collect_list_nat(Q)) ).

% Collect_conj_eq
tff(fact_4775_Collect__conj__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dl(P),Q)) = inf_inf_set_set_nat(collect_set_nat(P),collect_set_nat(Q)) ).

% Collect_conj_eq
tff(fact_4776_Collect__conj__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dm(P),Q)) = inf_inf_set_int(collect_int(P),collect_int(Q)) ).

% Collect_conj_eq
tff(fact_4777_Collect__conj__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dn(P),Q)) = inf_inf_set_nat(collect_nat(P),collect_nat(Q)) ).

% Collect_conj_eq
tff(fact_4778_Int__emptyI,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => ~ aa_set_complex_o(member_complex(X3),B2) )
     => ( inf_inf_set_complex(A2,B2) = bot_bot_set_complex ) ) ).

% Int_emptyI
tff(fact_4779_Int__emptyI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A2)
         => ~ aa_set_set_nat_o(member_set_nat(X3),B2) )
     => ( inf_inf_set_set_nat(A2,B2) = bot_bot_set_set_nat ) ) ).

% Int_emptyI
tff(fact_4780_Int__emptyI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => ~ aa_set_real_o(member_real(X3),B2) )
     => ( inf_inf_set_real(A2,B2) = bot_bot_set_real ) ) ).

% Int_emptyI
tff(fact_4781_Int__emptyI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A2)
         => ~ aa_set_o_o(member_o((X3)),B2) )
     => ( inf_inf_set_o(A2,B2) = bot_bot_set_o ) ) ).

% Int_emptyI
tff(fact_4782_Int__emptyI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => ~ aa_set_nat_o(member_nat(X3),B2) )
     => ( inf_inf_set_nat(A2,B2) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_4783_Int__emptyI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => ~ aa_set_int_o(member_int(X3),B2) )
     => ( inf_inf_set_int(A2,B2) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_4784_disjoint__iff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( inf_inf_set_complex(A2,B2) = bot_bot_set_complex )
    <=> ! [X4: complex] :
          ( aa_set_complex_o(member_complex(X4),A2)
         => ~ aa_set_complex_o(member_complex(X4),B2) ) ) ).

% disjoint_iff
tff(fact_4785_disjoint__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( inf_inf_set_set_nat(A2,B2) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),A2)
         => ~ aa_set_set_nat_o(member_set_nat(X4),B2) ) ) ).

% disjoint_iff
tff(fact_4786_disjoint__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A2)
         => ~ aa_set_real_o(member_real(X4),B2) ) ) ).

% disjoint_iff
tff(fact_4787_disjoint__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A2)
         => ~ aa_set_o_o(member_o((X4)),B2) ) ) ).

% disjoint_iff
tff(fact_4788_disjoint__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat(A2,B2) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A2)
         => ~ aa_set_nat_o(member_nat(X4),B2) ) ) ).

% disjoint_iff
tff(fact_4789_disjoint__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(A2,B2) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => ~ aa_set_int_o(member_int(X4),B2) ) ) ).

% disjoint_iff
tff(fact_4790_Int__empty__left,axiom,
    ! [B2: set_real] : inf_inf_set_real(bot_bot_set_real,B2) = bot_bot_set_real ).

% Int_empty_left
tff(fact_4791_Int__empty__left,axiom,
    ! [B2: set_o] : inf_inf_set_o(bot_bot_set_o,B2) = bot_bot_set_o ).

% Int_empty_left
tff(fact_4792_Int__empty__left,axiom,
    ! [B2: set_nat] : inf_inf_set_nat(bot_bot_set_nat,B2) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_4793_Int__empty__left,axiom,
    ! [B2: set_int] : inf_inf_set_int(bot_bot_set_int,B2) = bot_bot_set_int ).

% Int_empty_left
tff(fact_4794_Int__empty__right,axiom,
    ! [A2: set_real] : inf_inf_set_real(A2,bot_bot_set_real) = bot_bot_set_real ).

% Int_empty_right
tff(fact_4795_Int__empty__right,axiom,
    ! [A2: set_o] : inf_inf_set_o(A2,bot_bot_set_o) = bot_bot_set_o ).

% Int_empty_right
tff(fact_4796_Int__empty__right,axiom,
    ! [A2: set_nat] : inf_inf_set_nat(A2,bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_4797_Int__empty__right,axiom,
    ! [A2: set_int] : inf_inf_set_int(A2,bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_4798_disjoint__iff__not__equal,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
    <=> ! [X4: real] :
          ( aa_set_real_o(member_real(X4),A2)
         => ! [Xa4: real] :
              ( aa_set_real_o(member_real(Xa4),B2)
             => ( X4 != Xa4 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4799_disjoint__iff__not__equal,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A2)
         => ! [Xa4: $o] :
              ( aa_set_o_o(member_o((Xa4)),B2)
             => ( (X4)
              <=> ~ (Xa4) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4800_disjoint__iff__not__equal,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat(A2,B2) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A2)
         => ! [Xa4: nat] :
              ( aa_set_nat_o(member_nat(Xa4),B2)
             => ( X4 != Xa4 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4801_disjoint__iff__not__equal,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(A2,B2) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A2)
         => ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),B2)
             => ( X4 != Xa4 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4802_Int__mono,axiom,
    ! [A2: set_nat,C5: set_nat,B2: set_nat,D4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),D4)
       => aa_set_nat_o(ord_less_eq_set_nat(inf_inf_set_nat(A2,B2)),inf_inf_set_nat(C5,D4)) ) ) ).

% Int_mono
tff(fact_4803_Int__lower1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(inf_inf_set_nat(A2,B2)),A2) ).

% Int_lower1
tff(fact_4804_Int__lower2,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(inf_inf_set_nat(A2,B2)),B2) ).

% Int_lower2
tff(fact_4805_Int__absorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( inf_inf_set_nat(A2,B2) = B2 ) ) ).

% Int_absorb1
tff(fact_4806_Int__absorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( inf_inf_set_nat(A2,B2) = A2 ) ) ).

% Int_absorb2
tff(fact_4807_Int__greatest,axiom,
    ! [C5: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C5),A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C5),B2)
       => aa_set_nat_o(ord_less_eq_set_nat(C5),inf_inf_set_nat(A2,B2)) ) ) ).

% Int_greatest
tff(fact_4808_Int__Collect__mono,axiom,
    ! [A2: set_complex,B2: set_complex,P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => ( aa_complex_o(P,X3)
             => aa_complex_o(Q,X3) ) )
       => aa_set_complex_o(ord_le211207098394363844omplex(inf_inf_set_complex(A2,collect_complex(P))),inf_inf_set_complex(B2,collect_complex(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4809_Int__Collect__mono,axiom,
    ! [A2: set_real,B2: set_real,P: fun_real_o,Q: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
           => ( aa_real_o(P,X3)
             => aa_real_o(Q,X3) ) )
       => aa_set_real_o(ord_less_eq_set_real(inf_inf_set_real(A2,collect_real(P))),inf_inf_set_real(B2,collect_real(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4810_Int__Collect__mono,axiom,
    ! [A2: set_list_nat,B2: set_list_nat,P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(A2,B2)
     => ( ! [X3: list_nat] :
            ( member_list_nat(X3,A2)
           => ( aa_list_nat_o(P,X3)
             => aa_list_nat_o(Q,X3) ) )
       => ord_le6045566169113846134st_nat(inf_inf_set_list_nat(A2,collect_list_nat(P)),inf_inf_set_list_nat(B2,collect_list_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4811_Int__Collect__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),B2)
     => ( ! [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A2)
           => ( aa_set_nat_o(P,X3)
             => aa_set_nat_o(Q,X3) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(inf_inf_set_set_nat(A2,collect_set_nat(P))),inf_inf_set_set_nat(B2,collect_set_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4812_Int__Collect__mono,axiom,
    ! [A2: set_int,B2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => ( aa_int_o(P,X3)
             => aa_int_o(Q,X3) ) )
       => aa_set_int_o(ord_less_eq_set_int(inf_inf_set_int(A2,collect_int(P))),inf_inf_set_int(B2,collect_int(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4813_Int__Collect__mono,axiom,
    ! [A2: set_nat,B2: set_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
           => ( aa_nat_o(P,X3)
             => aa_nat_o(Q,X3) ) )
       => aa_set_nat_o(ord_less_eq_set_nat(inf_inf_set_nat(A2,collect_nat(P))),inf_inf_set_nat(B2,collect_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_4814_Int__insert__right,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      inf_inf_set_o(A2,insert_o((A),B2)) = $ite(aa_set_o_o(member_o((A)),A2),insert_o((A),inf_inf_set_o(A2,B2)),inf_inf_set_o(A2,B2)) ).

% Int_insert_right
tff(fact_4815_Int__insert__right,axiom,
    ! [A2: set_complex,A: complex,B2: set_complex] :
      inf_inf_set_complex(A2,insert_complex(A,B2)) = $ite(aa_set_complex_o(member_complex(A),A2),insert_complex(A,inf_inf_set_complex(A2,B2)),inf_inf_set_complex(A2,B2)) ).

% Int_insert_right
tff(fact_4816_Int__insert__right,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      inf_inf_set_real(A2,insert_real(A,B2)) = $ite(aa_set_real_o(member_real(A),A2),insert_real(A,inf_inf_set_real(A2,B2)),inf_inf_set_real(A2,B2)) ).

% Int_insert_right
tff(fact_4817_Int__insert__right,axiom,
    ! [A2: set_set_nat,A: set_nat,B2: set_set_nat] :
      inf_inf_set_set_nat(A2,insert_set_nat(A,B2)) = $ite(aa_set_set_nat_o(member_set_nat(A),A2),insert_set_nat(A,inf_inf_set_set_nat(A2,B2)),inf_inf_set_set_nat(A2,B2)) ).

% Int_insert_right
tff(fact_4818_Int__insert__right,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      inf_inf_set_int(A2,insert_int(A,B2)) = $ite(aa_set_int_o(member_int(A),A2),insert_int(A,inf_inf_set_int(A2,B2)),inf_inf_set_int(A2,B2)) ).

% Int_insert_right
tff(fact_4819_Int__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      inf_inf_set_nat(A2,insert_nat(A,B2)) = $ite(aa_set_nat_o(member_nat(A),A2),insert_nat(A,inf_inf_set_nat(A2,B2)),inf_inf_set_nat(A2,B2)) ).

% Int_insert_right
tff(fact_4820_Int__insert__left,axiom,
    ! [A: $o,B2: set_o,C5: set_o] :
      inf_inf_set_o(insert_o((A),B2),C5) = $ite(aa_set_o_o(member_o((A)),C5),insert_o((A),inf_inf_set_o(B2,C5)),inf_inf_set_o(B2,C5)) ).

% Int_insert_left
tff(fact_4821_Int__insert__left,axiom,
    ! [A: complex,B2: set_complex,C5: set_complex] :
      inf_inf_set_complex(insert_complex(A,B2),C5) = $ite(aa_set_complex_o(member_complex(A),C5),insert_complex(A,inf_inf_set_complex(B2,C5)),inf_inf_set_complex(B2,C5)) ).

% Int_insert_left
tff(fact_4822_Int__insert__left,axiom,
    ! [A: real,B2: set_real,C5: set_real] :
      inf_inf_set_real(insert_real(A,B2),C5) = $ite(aa_set_real_o(member_real(A),C5),insert_real(A,inf_inf_set_real(B2,C5)),inf_inf_set_real(B2,C5)) ).

% Int_insert_left
tff(fact_4823_Int__insert__left,axiom,
    ! [A: set_nat,B2: set_set_nat,C5: set_set_nat] :
      inf_inf_set_set_nat(insert_set_nat(A,B2),C5) = $ite(aa_set_set_nat_o(member_set_nat(A),C5),insert_set_nat(A,inf_inf_set_set_nat(B2,C5)),inf_inf_set_set_nat(B2,C5)) ).

% Int_insert_left
tff(fact_4824_Int__insert__left,axiom,
    ! [A: int,B2: set_int,C5: set_int] :
      inf_inf_set_int(insert_int(A,B2),C5) = $ite(aa_set_int_o(member_int(A),C5),insert_int(A,inf_inf_set_int(B2,C5)),inf_inf_set_int(B2,C5)) ).

% Int_insert_left
tff(fact_4825_Int__insert__left,axiom,
    ! [A: nat,B2: set_nat,C5: set_nat] :
      inf_inf_set_nat(insert_nat(A,B2),C5) = $ite(aa_set_nat_o(member_nat(A),C5),insert_nat(A,inf_inf_set_nat(B2,C5)),inf_inf_set_nat(B2,C5)) ).

% Int_insert_left
tff(fact_4826_Un__Int__distrib2,axiom,
    ! [B2: set_nat,C5: set_nat,A2: set_nat] : sup_sup_set_nat(inf_inf_set_nat(B2,C5),A2) = inf_inf_set_nat(sup_sup_set_nat(B2,A2),sup_sup_set_nat(C5,A2)) ).

% Un_Int_distrib2
tff(fact_4827_Int__Un__distrib2,axiom,
    ! [B2: set_nat,C5: set_nat,A2: set_nat] : inf_inf_set_nat(sup_sup_set_nat(B2,C5),A2) = sup_sup_set_nat(inf_inf_set_nat(B2,A2),inf_inf_set_nat(C5,A2)) ).

% Int_Un_distrib2
tff(fact_4828_Un__Int__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : sup_sup_set_nat(A2,inf_inf_set_nat(B2,C5)) = inf_inf_set_nat(sup_sup_set_nat(A2,B2),sup_sup_set_nat(A2,C5)) ).

% Un_Int_distrib
tff(fact_4829_Int__Un__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : inf_inf_set_nat(A2,sup_sup_set_nat(B2,C5)) = sup_sup_set_nat(inf_inf_set_nat(A2,B2),inf_inf_set_nat(A2,C5)) ).

% Int_Un_distrib
tff(fact_4830_Un__Int__crazy,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : sup_sup_set_nat(sup_sup_set_nat(inf_inf_set_nat(A2,B2),inf_inf_set_nat(B2,C5)),inf_inf_set_nat(C5,A2)) = inf_inf_set_nat(inf_inf_set_nat(sup_sup_set_nat(A2,B2),sup_sup_set_nat(B2,C5)),sup_sup_set_nat(C5,A2)) ).

% Un_Int_crazy
tff(fact_4831_Diff__Int__distrib2,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : inf_inf_set_nat(minus_minus_set_nat(A2,B2),C5) = minus_minus_set_nat(inf_inf_set_nat(A2,C5),inf_inf_set_nat(B2,C5)) ).

% Diff_Int_distrib2
tff(fact_4832_Diff__Int__distrib,axiom,
    ! [C5: set_nat,A2: set_nat,B2: set_nat] : inf_inf_set_nat(C5,minus_minus_set_nat(A2,B2)) = minus_minus_set_nat(inf_inf_set_nat(C5,A2),inf_inf_set_nat(C5,B2)) ).

% Diff_Int_distrib
tff(fact_4833_Diff__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,minus_minus_set_nat(A2,B2)) = inf_inf_set_nat(A2,B2) ).

% Diff_Diff_Int
tff(fact_4834_Diff__Int2,axiom,
    ! [A2: set_nat,C5: set_nat,B2: set_nat] : minus_minus_set_nat(inf_inf_set_nat(A2,C5),inf_inf_set_nat(B2,C5)) = minus_minus_set_nat(inf_inf_set_nat(A2,C5),B2) ).

% Diff_Int2
tff(fact_4835_Int__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : minus_minus_set_nat(inf_inf_set_nat(A2,B2),C5) = inf_inf_set_nat(A2,minus_minus_set_nat(B2,C5)) ).

% Int_Diff
tff(fact_4836_Int__Diff__disjoint,axiom,
    ! [A2: set_real,B2: set_real] : inf_inf_set_real(inf_inf_set_real(A2,B2),minus_minus_set_real(A2,B2)) = bot_bot_set_real ).

% Int_Diff_disjoint
tff(fact_4837_Int__Diff__disjoint,axiom,
    ! [A2: set_o,B2: set_o] : inf_inf_set_o(inf_inf_set_o(A2,B2),minus_minus_set_o(A2,B2)) = bot_bot_set_o ).

% Int_Diff_disjoint
tff(fact_4838_Int__Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] : inf_inf_set_int(inf_inf_set_int(A2,B2),minus_minus_set_int(A2,B2)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_4839_Int__Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : inf_inf_set_nat(inf_inf_set_nat(A2,B2),minus_minus_set_nat(A2,B2)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_4840_Diff__triv,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
     => ( minus_minus_set_real(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_4841_Diff__triv,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
     => ( minus_minus_set_o(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_4842_Diff__triv,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(A2,B2) = bot_bot_set_int )
     => ( minus_minus_set_int(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_4843_Diff__triv,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat(A2,B2) = bot_bot_set_nat )
     => ( minus_minus_set_nat(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_4844_Un__Int__assoc__eq,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( sup_sup_set_nat(inf_inf_set_nat(A2,B2),C5) = inf_inf_set_nat(A2,sup_sup_set_nat(B2,C5)) )
    <=> aa_set_nat_o(ord_less_eq_set_nat(C5),A2) ) ).

% Un_Int_assoc_eq
tff(fact_4845_Un__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : sup_sup_set_nat(minus_minus_set_nat(A2,B2),inf_inf_set_nat(A2,B2)) = A2 ).

% Un_Diff_Int
tff(fact_4846_Int__Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : sup_sup_set_nat(inf_inf_set_nat(A2,B2),minus_minus_set_nat(A2,B2)) = A2 ).

% Int_Diff_Un
tff(fact_4847_Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : minus_minus_set_nat(A2,inf_inf_set_nat(B2,C5)) = sup_sup_set_nat(minus_minus_set_nat(A2,B2),minus_minus_set_nat(A2,C5)) ).

% Diff_Int
tff(fact_4848_Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] : minus_minus_set_nat(A2,sup_sup_set_nat(B2,C5)) = inf_inf_set_nat(minus_minus_set_nat(A2,B2),minus_minus_set_nat(A2,C5)) ).

% Diff_Un
tff(fact_4849_vebt__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B4),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2,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)),TreeList2,Summary2),Xaa))
                 => ~ $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4850_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4851_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B4),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4852_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xaa))
                   => ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xaa))
                     => ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4853_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xaa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi2 )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xaa)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4854_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xaa))
               => ~ ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xaa))
                 => ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4855_max__bot2,axiom,
    ! [Xa: set_real] : ord_max_set_real(Xa,bot_bot_set_real) = Xa ).

% max_bot2
tff(fact_4856_max__bot2,axiom,
    ! [Xa: set_o] : ord_max_set_o(Xa,bot_bot_set_o) = Xa ).

% max_bot2
tff(fact_4857_max__bot2,axiom,
    ! [Xa: set_nat] : ord_max_set_nat(Xa,bot_bot_set_nat) = Xa ).

% max_bot2
tff(fact_4858_max__bot2,axiom,
    ! [Xa: set_int] : ord_max_set_int(Xa,bot_bot_set_int) = Xa ).

% max_bot2
tff(fact_4859_max__bot2,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),bot_bot_nat) = Xa ).

% max_bot2
tff(fact_4860_max__bot2,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xa),bot_bo4199563552545308370d_enat) = Xa ).

% max_bot2
tff(fact_4861_max__bot,axiom,
    ! [Xa: set_real] : ord_max_set_real(bot_bot_set_real,Xa) = Xa ).

% max_bot
tff(fact_4862_max__bot,axiom,
    ! [Xa: set_o] : ord_max_set_o(bot_bot_set_o,Xa) = Xa ).

% max_bot
tff(fact_4863_max__bot,axiom,
    ! [Xa: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xa) = Xa ).

% max_bot
tff(fact_4864_max__bot,axiom,
    ! [Xa: set_int] : ord_max_set_int(bot_bot_set_int,Xa) = Xa ).

% max_bot
tff(fact_4865_max__bot,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xa) = Xa ).

% max_bot
tff(fact_4866_max__bot,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,bot_bo4199563552545308370d_enat),Xa) = Xa ).

% max_bot
tff(fact_4867_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb3
tff(fact_4868_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_4869_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb3
tff(fact_4870_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_4871_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_4872_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb3
tff(fact_4873_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb3
tff(fact_4874_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb4
tff(fact_4875_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_4876_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb4
tff(fact_4877_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_4878_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_4879_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb4
tff(fact_4880_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb4
tff(fact_4881_inf__bot__left,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,bot_bo4199563552545308370d_enat),Xa) = bot_bo4199563552545308370d_enat ).

% inf_bot_left
tff(fact_4882_inf__bot__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(bot_bot_set_real,Xa) = bot_bot_set_real ).

% inf_bot_left
tff(fact_4883_inf__bot__left,axiom,
    ! [Xa: set_o] : inf_inf_set_o(bot_bot_set_o,Xa) = bot_bot_set_o ).

% inf_bot_left
tff(fact_4884_inf__bot__left,axiom,
    ! [Xa: set_nat] : inf_inf_set_nat(bot_bot_set_nat,Xa) = bot_bot_set_nat ).

% inf_bot_left
tff(fact_4885_inf__bot__left,axiom,
    ! [Xa: set_int] : inf_inf_set_int(bot_bot_set_int,Xa) = bot_bot_set_int ).

% inf_bot_left
tff(fact_4886_inf__bot__right,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,Xa),bot_bo4199563552545308370d_enat) = bot_bo4199563552545308370d_enat ).

% inf_bot_right
tff(fact_4887_inf__bot__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,bot_bot_set_real) = bot_bot_set_real ).

% inf_bot_right
tff(fact_4888_inf__bot__right,axiom,
    ! [Xa: set_o] : inf_inf_set_o(Xa,bot_bot_set_o) = bot_bot_set_o ).

% inf_bot_right
tff(fact_4889_inf__bot__right,axiom,
    ! [Xa: set_nat] : inf_inf_set_nat(Xa,bot_bot_set_nat) = bot_bot_set_nat ).

% inf_bot_right
tff(fact_4890_inf__bot__right,axiom,
    ! [Xa: set_int] : inf_inf_set_int(Xa,bot_bot_set_int) = bot_bot_set_int ).

% inf_bot_right
tff(fact_4891_sup__bot__left,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,bot_bo4199563552545308370d_enat),Xa) = Xa ).

% sup_bot_left
tff(fact_4892_sup__bot__left,axiom,
    ! [Xa: set_real] : sup_sup_set_real(bot_bot_set_real,Xa) = Xa ).

% sup_bot_left
tff(fact_4893_sup__bot__left,axiom,
    ! [Xa: set_o] : sup_sup_set_o(bot_bot_set_o,Xa) = Xa ).

% sup_bot_left
tff(fact_4894_sup__bot__left,axiom,
    ! [Xa: set_nat] : sup_sup_set_nat(bot_bot_set_nat,Xa) = Xa ).

% sup_bot_left
tff(fact_4895_sup__bot__left,axiom,
    ! [Xa: set_int] : sup_sup_set_int(bot_bot_set_int,Xa) = Xa ).

% sup_bot_left
tff(fact_4896_sup__bot__right,axiom,
    ! [Xa: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),bot_bo4199563552545308370d_enat) = Xa ).

% sup_bot_right
tff(fact_4897_sup__bot__right,axiom,
    ! [Xa: set_real] : sup_sup_set_real(Xa,bot_bot_set_real) = Xa ).

% sup_bot_right
tff(fact_4898_sup__bot__right,axiom,
    ! [Xa: set_o] : sup_sup_set_o(Xa,bot_bot_set_o) = Xa ).

% sup_bot_right
tff(fact_4899_sup__bot__right,axiom,
    ! [Xa: set_nat] : sup_sup_set_nat(Xa,bot_bot_set_nat) = Xa ).

% sup_bot_right
tff(fact_4900_sup__bot__right,axiom,
    ! [Xa: set_int] : sup_sup_set_int(Xa,bot_bot_set_int) = Xa ).

% sup_bot_right
tff(fact_4901_bot__eq__sup__iff,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ( bot_bo4199563552545308370d_enat = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) )
    <=> ( ( Xa = bot_bo4199563552545308370d_enat )
        & ( Y = bot_bo4199563552545308370d_enat ) ) ) ).

% bot_eq_sup_iff
tff(fact_4902_bot__eq__sup__iff,axiom,
    ! [Xa: set_real,Y: set_real] :
      ( ( bot_bot_set_real = sup_sup_set_real(Xa,Y) )
    <=> ( ( Xa = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
tff(fact_4903_bot__eq__sup__iff,axiom,
    ! [Xa: set_o,Y: set_o] :
      ( ( bot_bot_set_o = sup_sup_set_o(Xa,Y) )
    <=> ( ( Xa = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
tff(fact_4904_bot__eq__sup__iff,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = sup_sup_set_nat(Xa,Y) )
    <=> ( ( Xa = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_4905_bot__eq__sup__iff,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( bot_bot_set_int = sup_sup_set_int(Xa,Y) )
    <=> ( ( Xa = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_4906_sup__eq__bot__iff,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,Xa),Y) = bot_bo4199563552545308370d_enat )
    <=> ( ( Xa = bot_bo4199563552545308370d_enat )
        & ( Y = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_eq_bot_iff
tff(fact_4907_sup__eq__bot__iff,axiom,
    ! [Xa: set_real,Y: set_real] :
      ( ( sup_sup_set_real(Xa,Y) = bot_bot_set_real )
    <=> ( ( Xa = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
tff(fact_4908_sup__eq__bot__iff,axiom,
    ! [Xa: set_o,Y: set_o] :
      ( ( sup_sup_set_o(Xa,Y) = bot_bot_set_o )
    <=> ( ( Xa = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
tff(fact_4909_sup__eq__bot__iff,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat(Xa,Y) = bot_bot_set_nat )
    <=> ( ( Xa = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_4910_sup__eq__bot__iff,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( sup_sup_set_int(Xa,Y) = bot_bot_set_int )
    <=> ( ( Xa = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_4911_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = bot_bo4199563552545308370d_enat )
    <=> ( ( A = bot_bo4199563552545308370d_enat )
        & ( B = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_4912_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( sup_sup_set_real(A,B) = bot_bot_set_real )
    <=> ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_4913_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( sup_sup_set_o(A,B) = bot_bot_set_o )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_4914_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat(A,B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_4915_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( sup_sup_set_int(A,B) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_4916_sup__bot_Oleft__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,bot_bo4199563552545308370d_enat),A) = A ).

% sup_bot.left_neutral
tff(fact_4917_sup__bot_Oleft__neutral,axiom,
    ! [A: set_real] : sup_sup_set_real(bot_bot_set_real,A) = A ).

% sup_bot.left_neutral
tff(fact_4918_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] : sup_sup_set_o(bot_bot_set_o,A) = A ).

% sup_bot.left_neutral
tff(fact_4919_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] : sup_sup_set_nat(bot_bot_set_nat,A) = A ).

% sup_bot.left_neutral
tff(fact_4920_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] : sup_sup_set_int(bot_bot_set_int,A) = A ).

% sup_bot.left_neutral
tff(fact_4921_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( bot_bo4199563552545308370d_enat = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) )
    <=> ( ( A = bot_bo4199563552545308370d_enat )
        & ( B = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_4922_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( bot_bot_set_real = sup_sup_set_real(A,B) )
    <=> ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_4923_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o = sup_sup_set_o(A,B) )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_4924_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = sup_sup_set_nat(A,B) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_4925_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int = sup_sup_set_int(A,B) )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_4926_sup__bot_Oright__neutral,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),bot_bo4199563552545308370d_enat) = A ).

% sup_bot.right_neutral
tff(fact_4927_sup__bot_Oright__neutral,axiom,
    ! [A: set_real] : sup_sup_set_real(A,bot_bot_set_real) = A ).

% sup_bot.right_neutral
tff(fact_4928_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] : sup_sup_set_o(A,bot_bot_set_o) = A ).

% sup_bot.right_neutral
tff(fact_4929_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] : sup_sup_set_nat(A,bot_bot_set_nat) = A ).

% sup_bot.right_neutral
tff(fact_4930_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] : sup_sup_set_int(A,bot_bot_set_int) = A ).

% sup_bot.right_neutral
tff(fact_4931_max__less__iff__conj,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ord_max_Code_integer(Xa,Y)),Z)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Z)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4932_max__less__iff__conj,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xa),Y)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4933_max__less__iff__conj,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Xa),Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4934_max__less__iff__conj,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Xa),Y)),Z)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4935_max__less__iff__conj,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4936_max__less__iff__conj,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(Xa),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4937_max__less__iff__conj,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xa),Y)),Z)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Z)
        & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4938_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_4939_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_4940_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bot_fun_real_o) ).

% bot_set_def
tff(fact_4941_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_4942_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_4943_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_4944_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_4945_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_4946_lt__ex,axiom,
    ! [Xa: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xa) ).

% lt_ex
tff(fact_4947_lt__ex,axiom,
    ! [Xa: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xa) ).

% lt_ex
tff(fact_4948_lt__ex,axiom,
    ! [Xa: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xa) ).

% lt_ex
tff(fact_4949_gt__ex,axiom,
    ! [Xa: real] :
    ? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X_13) ).

% gt_ex
tff(fact_4950_gt__ex,axiom,
    ! [Xa: rat] :
    ? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X_13) ).

% gt_ex
tff(fact_4951_gt__ex,axiom,
    ! [Xa: nat] :
    ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X_13) ).

% gt_ex
tff(fact_4952_gt__ex,axiom,
    ! [Xa: int] :
    ? [X_13: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X_13) ).

% gt_ex
tff(fact_4953_dense,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ? [Z4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z4)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),Y) ) ) ).

% dense
tff(fact_4954_dense,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ? [Z4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z4)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),Y) ) ) ).

% dense
tff(fact_4955_less__imp__neq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_4956_less__imp__neq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_4957_less__imp__neq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_4958_less__imp__neq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_4959_less__imp__neq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_4960_less__imp__neq,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_4961_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order.asym
tff(fact_4962_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order.asym
tff(fact_4963_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_4964_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_4965_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_4966_order_Oasym,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A) ) ).

% order.asym
tff(fact_4967_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4968_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4969_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4970_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4971_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4972_ord__eq__less__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A = B )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4973_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( B = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4974_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4975_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4976_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4977_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4978_ord__less__eq__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( ( B = C )
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4979_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X3)
             => aa_nat_o(P,Y4) )
         => aa_nat_o(P,X3) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_4980_less__induct,axiom,
    ! [P: fun_Extended_enat_o,A: extended_enat] :
      ( ! [X3: extended_enat] :
          ( ! [Y4: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y4),X3)
             => aa_Extended_enat_o(P,Y4) )
         => aa_Extended_enat_o(P,X3) )
     => aa_Extended_enat_o(P,A) ) ).

% less_induct
tff(fact_4981_antisym__conv3,axiom,
    ! [Y: real,Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_4982_antisym__conv3,axiom,
    ! [Y: rat,Xa: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_4983_antisym__conv3,axiom,
    ! [Y: num,Xa: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_4984_antisym__conv3,axiom,
    ! [Y: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_4985_antisym__conv3,axiom,
    ! [Y: int,Xa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_4986_antisym__conv3,axiom,
    ! [Y: extended_enat,Xa: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa)
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_4987_linorder__cases,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( ( Xa != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_cases
tff(fact_4988_linorder__cases,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( ( Xa != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_cases
tff(fact_4989_linorder__cases,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( ( Xa != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_cases
tff(fact_4990_linorder__cases,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( ( Xa != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_cases
tff(fact_4991_linorder__cases,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( ( Xa != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_cases
tff(fact_4992_linorder__cases,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ( ( Xa != Y )
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ) ).

% linorder_cases
tff(fact_4993_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% dual_order.asym
tff(fact_4994_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% dual_order.asym
tff(fact_4995_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_4996_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_4997_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_4998_dual__order_Oasym,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ).

% dual_order.asym
tff(fact_4999_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_5000_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_5001_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_5002_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_5003_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_5004_dual__order_Oirrefl,axiom,
    ! [A: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),A) ).

% dual_order.irrefl
tff(fact_5005_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
    <=> ? [N4: nat] :
          ( aa_nat_o(P,N4)
          & ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N4)
             => ~ aa_nat_o(P,M3) ) ) ) ).

% exists_least_iff
tff(fact_5006_exists__least__iff,axiom,
    ! [P: fun_Extended_enat_o] :
      ( ? [X_1: extended_enat] : aa_Extended_enat_o(P,X_1)
    <=> ? [N4: extended_enat] :
          ( aa_Extended_enat_o(P,N4)
          & ! [M3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,M3),N4)
             => ~ aa_Extended_enat_o(P,M3) ) ) ) ).

% exists_least_iff
tff(fact_5007_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A: real,B: real] :
      ( ! [A4: real,B4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B4)
         => aa_real_o(aa_real_fun_real_o(P,A4),B4) )
     => ( ! [A4: real] : aa_real_o(aa_real_fun_real_o(P,A4),A4)
       => ( ! [A4: real,B4: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B4),A4)
             => aa_real_o(aa_real_fun_real_o(P,A4),B4) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_5008_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B4)
         => aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
     => ( ! [A4: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A4),A4)
       => ( ! [A4: rat,B4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B4),A4)
             => aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_5009_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A4: num,B4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B4)
         => aa_num_o(aa_num_fun_num_o(P,A4),B4) )
     => ( ! [A4: num] : aa_num_o(aa_num_fun_num_o(P,A4),A4)
       => ( ! [A4: num,B4: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B4),A4)
             => aa_num_o(aa_num_fun_num_o(P,A4),B4) )
         => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_5010_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
         => aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),A4)
       => ( ! [A4: nat,B4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B4),A4)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_5011_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
         => aa_int_o(aa_int_fun_int_o(P,A4),B4) )
     => ( ! [A4: int] : aa_int_o(aa_int_fun_int_o(P,A4),A4)
       => ( ! [A4: int,B4: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B4),A4)
             => aa_int_o(aa_int_fun_int_o(P,A4),B4) )
         => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_5012_linorder__less__wlog,axiom,
    ! [P: fun_Ex239764483819389678enat_o,A: extended_enat,B: extended_enat] :
      ( ! [A4: extended_enat,B4: extended_enat] :
          ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A4),B4)
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,A4),B4) )
     => ( ! [A4: extended_enat] : aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,A4),A4)
       => ( ! [A4: extended_enat,B4: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,B4),A4)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,A4),B4) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_5013_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans
tff(fact_5014_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans
tff(fact_5015_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans
tff(fact_5016_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans
tff(fact_5017_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans
tff(fact_5018_order_Ostrict__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% order.strict_trans
tff(fact_5019_not__less__iff__gr__or__eq,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_5020_not__less__iff__gr__or__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_5021_not__less__iff__gr__or__eq,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_5022_not__less__iff__gr__or__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_5023_not__less__iff__gr__or__eq,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_5024_not__less__iff__gr__or__eq,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_5025_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_5026_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_5027_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_5028_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_5029_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_5030_dual__order_Ostrict__trans,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_5031_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_5032_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_5033_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_5034_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_5035_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_5036_order_Ostrict__implies__not__eq,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_5037_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_5038_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_5039_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_5040_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_5041_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_5042_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_5043_linorder__neqE,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_5044_linorder__neqE,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_5045_linorder__neqE,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_5046_linorder__neqE,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_5047_linorder__neqE,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_5048_linorder__neqE,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ( Xa != Y )
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_5049_order__less__asym,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_asym
tff(fact_5050_order__less__asym,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_asym
tff(fact_5051_order__less__asym,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_asym
tff(fact_5052_order__less__asym,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_asym
tff(fact_5053_order__less__asym,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_asym
tff(fact_5054_order__less__asym,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ).

% order_less_asym
tff(fact_5055_linorder__neq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_5056_linorder__neq__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_5057_linorder__neq__iff,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_5058_linorder__neq__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_5059_linorder__neq__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_5060_linorder__neq__iff,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ( Xa != Y )
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
        | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_5061_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order_less_asym'
tff(fact_5062_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order_less_asym'
tff(fact_5063_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_5064_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_5065_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_5066_order__less__asym_H,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A) ) ).

% order_less_asym'
tff(fact_5067_order__less__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_less_trans
tff(fact_5068_order__less__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_less_trans
tff(fact_5069_order__less__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_less_trans
tff(fact_5070_order__less__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_less_trans
tff(fact_5071_order__less__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_less_trans
tff(fact_5072_order__less__trans,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Z)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Z) ) ) ).

% order_less_trans
tff(fact_5073_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5074_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5075_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5076_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5077_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5078_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B: real,C: real] :
      ( ( A = aa_rea1194391308913012860d_enat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X3)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5079_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5080_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5081_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5082_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_5083_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5084_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5085_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5086_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5087_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5088_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_rea1194391308913012860d_enat(F,B) = C )
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X3)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5089_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5090_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5091_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5092_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_5093_order__less__irrefl,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Xa) ).

% order_less_irrefl
tff(fact_5094_order__less__irrefl,axiom,
    ! [Xa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Xa) ).

% order_less_irrefl
tff(fact_5095_order__less__irrefl,axiom,
    ! [Xa: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Xa) ).

% order_less_irrefl
tff(fact_5096_order__less__irrefl,axiom,
    ! [Xa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Xa) ).

% order_less_irrefl
tff(fact_5097_order__less__irrefl,axiom,
    ! [Xa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xa) ).

% order_less_irrefl
tff(fact_5098_order__less__irrefl,axiom,
    ! [Xa: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Xa) ).

% order_less_irrefl
tff(fact_5099_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5100_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5101_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5102_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5103_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5104_order__less__subst1,axiom,
    ! [A: real,F: fun_Ex1451620480498224165t_real,B: extended_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_Ext2541572767299617110t_real(F,B))
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => ( ! [X3: extended_enat,Y3: extended_enat] :
              ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Ext2541572767299617110t_real(F,X3)),aa_Ext2541572767299617110t_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_Ext2541572767299617110t_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5105_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5106_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5107_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5108_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_5109_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5110_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5111_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5112_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5113_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5114_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,B)),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X3)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5115_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5116_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5117_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5118_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_5119_order__less__not__sym,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_not_sym
tff(fact_5120_order__less__not__sym,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_not_sym
tff(fact_5121_order__less__not__sym,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_not_sym
tff(fact_5122_order__less__not__sym,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_not_sym
tff(fact_5123_order__less__not__sym,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_not_sym
tff(fact_5124_order__less__not__sym,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ).

% order_less_not_sym
tff(fact_5125_order__less__imp__triv,axiom,
    ! [Xa: real,Y: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_5126_order__less__imp__triv,axiom,
    ! [Xa: rat,Y: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_5127_order__less__imp__triv,axiom,
    ! [Xa: num,Y: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_5128_order__less__imp__triv,axiom,
    ! [Xa: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_5129_order__less__imp__triv,axiom,
    ! [Xa: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_5130_order__less__imp__triv,axiom,
    ! [Xa: extended_enat,Y: extended_enat,P: $o] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_5131_linorder__less__linear,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      | ( Xa = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_less_linear
tff(fact_5132_linorder__less__linear,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      | ( Xa = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_less_linear
tff(fact_5133_linorder__less__linear,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      | ( Xa = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_less_linear
tff(fact_5134_linorder__less__linear,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      | ( Xa = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_less_linear
tff(fact_5135_linorder__less__linear,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      | ( Xa = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_less_linear
tff(fact_5136_linorder__less__linear,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
      | ( Xa = Y )
      | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ).

% linorder_less_linear
tff(fact_5137_order__less__imp__not__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_5138_order__less__imp__not__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_5139_order__less__imp__not__eq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_5140_order__less__imp__not__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_5141_order__less__imp__not__eq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_5142_order__less__imp__not__eq,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_5143_order__less__imp__not__eq2,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_5144_order__less__imp__not__eq2,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_5145_order__less__imp__not__eq2,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_5146_order__less__imp__not__eq2,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_5147_order__less__imp__not__eq2,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_5148_order__less__imp__not__eq2,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_5149_order__less__imp__not__less,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_5150_order__less__imp__not__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_5151_order__less__imp__not__less,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_5152_order__less__imp__not__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_5153_order__less__imp__not__less,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_5154_order__less__imp__not__less,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_5155_leD,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% leD
tff(fact_5156_leD,axiom,
    ! [Y: extended_enat,Xa: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xa)
     => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y) ) ).

% leD
tff(fact_5157_leD,axiom,
    ! [Y: set_nat,Xa: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),Xa)
     => ~ aa_set_nat_o(ord_less_set_nat(Xa),Y) ) ).

% leD
tff(fact_5158_leD,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% leD
tff(fact_5159_leD,axiom,
    ! [Y: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y) ) ).

% leD
tff(fact_5160_leD,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ).

% leD
tff(fact_5161_leD,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ).

% leD
tff(fact_5162_leI,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ).

% leI
tff(fact_5163_leI,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xa) ) ).

% leI
tff(fact_5164_leI,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% leI
tff(fact_5165_leI,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% leI
tff(fact_5166_leI,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% leI
tff(fact_5167_leI,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% leI
tff(fact_5168_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_5169_nless__le,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
    <=> ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_5170_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_5171_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_5172_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_5173_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_5174_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_5175_antisym__conv1,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_5176_antisym__conv1,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_5177_antisym__conv1,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_5178_antisym__conv1,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_5179_antisym__conv1,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_5180_antisym__conv1,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_5181_antisym__conv1,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_5182_antisym__conv2,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_5183_antisym__conv2,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
     => ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_5184_antisym__conv2,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( ~ aa_set_nat_o(ord_less_set_nat(Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_5185_antisym__conv2,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_5186_antisym__conv2,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_5187_antisym__conv2,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_5188_antisym__conv2,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_5189_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X3) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_ge
tff(fact_5190_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_5191_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_le
tff(fact_5192_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_5193_less__le__not__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_5194_less__le__not__le,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
        & ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_5195_less__le__not__le,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) ) ) ).

% less_le_not_le
tff(fact_5196_less__le__not__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_5197_less__le__not__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_5198_less__le__not__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_5199_less__le__not__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_5200_not__le__imp__less,axiom,
    ! [Y: real,Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% not_le_imp_less
tff(fact_5201_not__le__imp__less,axiom,
    ! [Y: extended_enat,Xa: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xa)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y) ) ).

% not_le_imp_less
tff(fact_5202_not__le__imp__less,axiom,
    ! [Y: rat,Xa: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% not_le_imp_less
tff(fact_5203_not__le__imp__less,axiom,
    ! [Y: num,Xa: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y) ) ).

% not_le_imp_less
tff(fact_5204_not__le__imp__less,axiom,
    ! [Y: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ).

% not_le_imp_less
tff(fact_5205_not__le__imp__less,axiom,
    ! [Y: int,Xa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ).

% not_le_imp_less
tff(fact_5206_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_5207_order_Oorder__iff__strict,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_5208_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_set_nat(A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_5209_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_5210_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_5211_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_5212_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_5213_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_5214_order_Ostrict__iff__order,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_5215_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_5216_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_5217_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_5218_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_5219_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_5220_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans1
tff(fact_5221_order_Ostrict__trans1,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% order.strict_trans1
tff(fact_5222_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% order.strict_trans1
tff(fact_5223_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans1
tff(fact_5224_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans1
tff(fact_5225_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_5226_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans1
tff(fact_5227_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans2
tff(fact_5228_order_Ostrict__trans2,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),C)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% order.strict_trans2
tff(fact_5229_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% order.strict_trans2
tff(fact_5230_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans2
tff(fact_5231_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans2
tff(fact_5232_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_5233_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans2
tff(fact_5234_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% order.strict_iff_not
tff(fact_5235_order_Ostrict__iff__not,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
        & ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_5236_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% order.strict_iff_not
tff(fact_5237_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_5238_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_5239_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_5240_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_5241_dense__ge__bounded,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xa)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Xa)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),W2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_5242_dense__ge__bounded,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xa)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Xa)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_5243_dense__le__bounded,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_5244_dense__le__bounded,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_5245_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5246_dual__order_Oorder__iff__strict,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5247_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_set_nat(B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5248_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5249_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5250_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5251_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5252_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5253_dual__order_Ostrict__iff__order,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5254_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5255_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5256_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5257_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5258_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5259_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_5260_dual__order_Ostrict__trans1,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_5261_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_set_nat(C),B)
       => aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_5262_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_5263_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_5264_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_5265_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_5266_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_5267_dual__order_Ostrict__trans2,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,C),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_5268_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
       => aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_5269_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_5270_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_5271_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_5272_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_5273_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_5274_dual__order_Ostrict__iff__not,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A)
        & ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_5275_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_5276_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_5277_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_5278_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_5279_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_5280_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% order.strict_implies_order
tff(fact_5281_order_Ostrict__implies__order,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B) ) ).

% order.strict_implies_order
tff(fact_5282_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% order.strict_implies_order
tff(fact_5283_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_5284_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_5285_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_5286_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_5287_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_5288_dual__order_Ostrict__implies__order,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_5289_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).

% dual_order.strict_implies_order
tff(fact_5290_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_5291_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_5292_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_5293_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_5294_order__le__less,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_5295_order__le__less,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_5296_order__le__less,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
    <=> ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_5297_order__le__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_5298_order__le__less,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_5299_order__le__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_5300_order__le__less,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_5301_order__less__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_5302_order__less__le,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_5303_order__less__le,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_5304_order__less__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_5305_order__less__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_5306_order__less__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_5307_order__less__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_5308_linorder__not__le,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_not_le
tff(fact_5309_linorder__not__le,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ).

% linorder_not_le
tff(fact_5310_linorder__not__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_not_le
tff(fact_5311_linorder__not__le,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_not_le
tff(fact_5312_linorder__not__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_not_le
tff(fact_5313_linorder__not__le,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_not_le
tff(fact_5314_linorder__not__less,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ).

% linorder_not_less
tff(fact_5315_linorder__not__less,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Xa) ) ).

% linorder_not_less
tff(fact_5316_linorder__not__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% linorder_not_less
tff(fact_5317_linorder__not__less,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% linorder_not_less
tff(fact_5318_linorder__not__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% linorder_not_less
tff(fact_5319_linorder__not__less,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% linorder_not_less
tff(fact_5320_order__less__imp__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% order_less_imp_le
tff(fact_5321_order__less__imp__le,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y) ) ).

% order_less_imp_le
tff(fact_5322_order__less__imp__le,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
     => aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) ) ).

% order_less_imp_le
tff(fact_5323_order__less__imp__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% order_less_imp_le
tff(fact_5324_order__less__imp__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ).

% order_less_imp_le
tff(fact_5325_order__less__imp__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ).

% order_less_imp_le
tff(fact_5326_order__less__imp__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ).

% order_less_imp_le
tff(fact_5327_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( A != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_le_neq_trans
tff(fact_5328_order__le__neq__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
     => ( ( A != B )
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_5329_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( ( A != B )
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% order_le_neq_trans
tff(fact_5330_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_5331_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_5332_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_5333_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_5334_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_neq_le_trans
tff(fact_5335_order__neq__le__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A != B )
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),B)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_5336_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% order_neq_le_trans
tff(fact_5337_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_5338_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_5339_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_5340_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_5341_order__le__less__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_5342_order__le__less__trans,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Z)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_5343_order__le__less__trans,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_set_nat(Y),Z)
       => aa_set_nat_o(ord_less_set_nat(Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_5344_order__le__less__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_5345_order__le__less__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_5346_order__le__less__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_5347_order__le__less__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_5348_order__less__le__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_5349_order__less__le__trans,axiom,
    ! [Xa: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Y),Z)
       => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_5350_order__less__le__trans,axiom,
    ! [Xa: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Y),Z)
       => aa_set_nat_o(ord_less_set_nat(Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_5351_order__less__le__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_5352_order__less__le__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_5353_order__less__le__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_5354_order__less__le__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_5355_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5356_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_re6065469302365115155d_enat,B: real,C: real] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_rea1194391308913012860d_enat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X3)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rea1194391308913012860d_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5357_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5358_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_ra2395272406506251367d_enat,B: rat,C: rat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_rat_Extended_enat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rat_Extended_enat(F,X3)),aa_rat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rat_Extended_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5359_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5360_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_nu2294053855564895909d_enat,B: num,C: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_num_Extended_enat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(F,X3)),aa_num_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_num_Extended_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5361_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5362_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_na8797796813480665199d_enat,B: nat,C: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_nat_Extended_enat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X3)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_nat_Extended_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5363_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5364_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: fun_in5914218760682845459d_enat,B: int,C: int] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,A),aa_int_Extended_enat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X3)),aa_int_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_int_Extended_enat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_5365_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5366_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_ra2395272406506251367d_enat,C: extended_enat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rat_Extended_enat(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rat_Extended_enat(F,X3)),aa_rat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rat_Extended_enat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5367_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5368_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5369_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5370_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5371_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X3)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5372_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_nu2294053855564895909d_enat,C: extended_enat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(F,B)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_num_Extended_enat(F,X3)),aa_num_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5373_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5374_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_5375_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5376_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: fun_ra2395272406506251367d_enat,B: rat,C: rat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rat_Extended_enat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rat_Extended_enat(F,X3)),aa_rat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_rat_Extended_enat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5377_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5378_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5379_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5380_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5381_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X3)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5382_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: fun_nu2294053855564895909d_enat,B: num,C: num] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_num_Extended_enat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_num_Extended_enat(F,X3)),aa_num_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_num_Extended_enat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5383_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5384_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_5385_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5386_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_re6065469302365115155d_enat,C: extended_enat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rea1194391308913012860d_enat(F,B)),C)
       => ( ! [X3: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,X3)),aa_rea1194391308913012860d_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rea1194391308913012860d_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5387_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5388_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_ra2395272406506251367d_enat,C: extended_enat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_rat_Extended_enat(F,B)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rat_Extended_enat(F,X3)),aa_rat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_rat_Extended_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5389_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5390_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_nu2294053855564895909d_enat,C: extended_enat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_num_Extended_enat(F,B)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(F,X3)),aa_num_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_num_Extended_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5391_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5392_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_na8797796813480665199d_enat,C: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_nat_Extended_enat(F,B)),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,X3)),aa_nat_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5393_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5394_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_in5914218760682845459d_enat,C: extended_enat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,aa_int_Extended_enat(F,B)),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,X3)),aa_int_Extended_enat(F,Y3)) )
         => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_int_Extended_enat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_5395_linorder__le__less__linear,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_5396_linorder__le__less__linear,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
      | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_5397_linorder__le__less__linear,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_5398_linorder__le__less__linear,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_5399_linorder__le__less__linear,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_5400_linorder__le__less__linear,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_5401_order__le__imp__less__or__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5402_order__le__imp__less__or__eq,axiom,
    ! [Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,Xa),Y)
     => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5403_order__le__imp__less__or__eq,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xa),Y)
     => ( aa_set_nat_o(ord_less_set_nat(Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5404_order__le__imp__less__or__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5405_order__le__imp__less__or__eq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5406_order__le__imp__less__or__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5407_order__le__imp__less__or__eq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5408_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_5409_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_5410_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_5411_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_5412_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_5413_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_5414_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_5415_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_5416_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_5417_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_5418_bot_Oextremum,axiom,
    ! [A: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A) ).

% bot.extremum
tff(fact_5419_bot_Oextremum,axiom,
    ! [A: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A) ).

% bot.extremum
tff(fact_5420_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% bot.extremum
tff(fact_5421_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_5422_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_5423_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ aa_set_real_o(ord_less_set_real(A),bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_5424_bot_Oextremum__strict,axiom,
    ! [A: set_o] : ~ aa_set_o_o(ord_less_set_o(A),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_5425_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_5426_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ aa_set_int_o(ord_less_set_int(A),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_5427_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_5428_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] : ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),bot_bo4199563552545308370d_enat) ).

% bot.extremum_strict
tff(fact_5429_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> aa_set_real_o(ord_less_set_real(bot_bot_set_real),A) ) ).

% bot.not_eq_extremum
tff(fact_5430_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
    <=> aa_set_o_o(ord_less_set_o(bot_bot_set_o),A) ) ).

% bot.not_eq_extremum
tff(fact_5431_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_set_nat(bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_5432_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),A) ) ).

% bot.not_eq_extremum
tff(fact_5433_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_5434_bot_Onot__eq__extremum,axiom,
    ! [A: extended_enat] :
      ( ( A != bot_bo4199563552545308370d_enat )
    <=> aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,bot_bo4199563552545308370d_enat),A) ) ).

% bot.not_eq_extremum
tff(fact_5435_less__infI1,axiom,
    ! [A: set_nat,Xa: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),Xa)
     => aa_set_nat_o(ord_less_set_nat(inf_inf_set_nat(A,B)),Xa) ) ).

% less_infI1
tff(fact_5436_less__infI1,axiom,
    ! [A: real,Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B)),Xa) ) ).

% less_infI1
tff(fact_5437_less__infI1,axiom,
    ! [A: rat,Xa: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inf_inf_rat(A),B)),Xa) ) ).

% less_infI1
tff(fact_5438_less__infI1,axiom,
    ! [A: nat,Xa: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),Xa) ) ).

% less_infI1
tff(fact_5439_less__infI1,axiom,
    ! [A: int,Xa: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(inf_inf_int(A),B)),Xa) ) ).

% less_infI1
tff(fact_5440_less__infI1,axiom,
    ! [A: extended_enat,Xa: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),Xa)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),Xa) ) ).

% less_infI1
tff(fact_5441_less__infI2,axiom,
    ! [B: set_nat,Xa: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),Xa)
     => aa_set_nat_o(ord_less_set_nat(inf_inf_set_nat(A,B)),Xa) ) ).

% less_infI2
tff(fact_5442_less__infI2,axiom,
    ! [B: real,Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B)),Xa) ) ).

% less_infI2
tff(fact_5443_less__infI2,axiom,
    ! [B: rat,Xa: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inf_inf_rat(A),B)),Xa) ) ).

% less_infI2
tff(fact_5444_less__infI2,axiom,
    ! [B: nat,Xa: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),Xa) ) ).

% less_infI2
tff(fact_5445_less__infI2,axiom,
    ! [B: int,Xa: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(inf_inf_int(A),B)),Xa) ) ).

% less_infI2
tff(fact_5446_less__infI2,axiom,
    ! [B: extended_enat,Xa: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),Xa)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),Xa) ) ).

% less_infI2
tff(fact_5447_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( inf_inf_set_nat(A,B) = A ) ) ).

% inf.absorb3
tff(fact_5448_inf_Oabsorb3,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5449_inf_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(inf_inf_rat(A),B) = A ) ) ).

% inf.absorb3
tff(fact_5450_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5451_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(inf_inf_int(A),B) = A ) ) ).

% inf.absorb3
tff(fact_5452_inf_Oabsorb3,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_5453_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => ( inf_inf_set_nat(A,B) = B ) ) ).

% inf.absorb4
tff(fact_5454_inf_Oabsorb4,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5455_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(inf_inf_rat(A),B) = B ) ) ).

% inf.absorb4
tff(fact_5456_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5457_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(inf_inf_int(A),B) = B ) ) ).

% inf.absorb4
tff(fact_5458_inf_Oabsorb4,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_5459_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),inf_inf_set_nat(B,C))
     => ~ ( aa_set_nat_o(ord_less_set_nat(A),B)
         => ~ aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5460_inf_Ostrict__boundedE,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,B),C))
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5461_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(inf_inf_rat(B),C))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5462_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5463_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(inf_inf_int(B),C))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5464_inf_Ostrict__boundedE,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,B),C))
     => ~ ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
         => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_5465_inf_Ostrict__order__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( ( A = inf_inf_set_nat(A,B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5466_inf_Ostrict__order__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5467_inf_Ostrict__order__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ( A = aa_rat_rat(inf_inf_rat(A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5468_inf_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5469_inf_Ostrict__order__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ( A = aa_int_int(inf_inf_int(A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5470_inf_Ostrict__order__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_5471_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),C)
     => aa_set_nat_o(ord_less_set_nat(inf_inf_set_nat(A,B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5472_inf_Ostrict__coboundedI1,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5473_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inf_inf_rat(A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5474_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5475_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(inf_inf_int(A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5476_inf_Ostrict__coboundedI1,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),C)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_5477_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),C)
     => aa_set_nat_o(ord_less_set_nat(inf_inf_set_nat(A,B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5478_inf_Ostrict__coboundedI2,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(inf_inf_real,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5479_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inf_inf_rat(A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5480_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5481_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(inf_inf_int(A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5482_inf_Ostrict__coboundedI2,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),C)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(inf_in1870772243966228564d_enat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_5483_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(C),B)
     => aa_set_nat_o(ord_less_set_nat(C),sup_sup_set_nat(A,B)) ) ).

% sup.strict_coboundedI2
tff(fact_5484_sup_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_5485_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(sup_sup_rat(A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_5486_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_5487_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(sup_sup_int(A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_5488_sup_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_5489_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(C),A)
     => aa_set_nat_o(ord_less_set_nat(C),sup_sup_set_nat(A,B)) ) ).

% sup.strict_coboundedI1
tff(fact_5490_sup_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_5491_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(sup_sup_rat(A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_5492_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_5493_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(sup_sup_int(A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_5494_sup_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_5495_sup_Ostrict__order__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( ( A = sup_sup_set_nat(A,B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5496_sup_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5497_sup_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(sup_sup_rat(A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5498_sup_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5499_sup_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(sup_sup_int(A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5500_sup_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_5501_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(sup_sup_set_nat(B,C)),A)
     => ~ ( aa_set_nat_o(ord_less_set_nat(B),A)
         => ~ aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% sup.strict_boundedE
tff(fact_5502_sup_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,B),C)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_5503_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(sup_sup_rat(B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_5504_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_5505_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(sup_sup_int(B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_5506_sup_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,B),C)),A)
     => ~ ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
         => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_5507_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( sup_sup_set_nat(A,B) = B ) ) ).

% sup.absorb4
tff(fact_5508_sup_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B) = B ) ) ).

% sup.absorb4
tff(fact_5509_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(sup_sup_rat(A),B) = B ) ) ).

% sup.absorb4
tff(fact_5510_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_5511_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(sup_sup_int(A),B) = B ) ) ).

% sup.absorb4
tff(fact_5512_sup_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_5513_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => ( sup_sup_set_nat(A,B) = A ) ) ).

% sup.absorb3
tff(fact_5514_sup_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B) = A ) ) ).

% sup.absorb3
tff(fact_5515_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(sup_sup_rat(A),B) = A ) ) ).

% sup.absorb3
tff(fact_5516_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_5517_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(sup_sup_int(A),B) = A ) ) ).

% sup.absorb3
tff(fact_5518_sup_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_5519_less__supI2,axiom,
    ! [Xa: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),B)
     => aa_set_nat_o(ord_less_set_nat(Xa),sup_sup_set_nat(A,B)) ) ).

% less_supI2
tff(fact_5520_less__supI2,axiom,
    ! [Xa: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B)) ) ).

% less_supI2
tff(fact_5521_less__supI2,axiom,
    ! [Xa: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(sup_sup_rat(A),B)) ) ).

% less_supI2
tff(fact_5522_less__supI2,axiom,
    ! [Xa: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI2
tff(fact_5523_less__supI2,axiom,
    ! [Xa: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(sup_sup_int(A),B)) ) ).

% less_supI2
tff(fact_5524_less__supI2,axiom,
    ! [Xa: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% less_supI2
tff(fact_5525_less__supI1,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xa),A)
     => aa_set_nat_o(ord_less_set_nat(Xa),sup_sup_set_nat(A,B)) ) ).

% less_supI1
tff(fact_5526_less__supI1,axiom,
    ! [Xa: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(sup_sup_real,A),B)) ) ).

% less_supI1
tff(fact_5527_less__supI1,axiom,
    ! [Xa: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(sup_sup_rat(A),B)) ) ).

% less_supI1
tff(fact_5528_less__supI1,axiom,
    ! [Xa: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI1
tff(fact_5529_less__supI1,axiom,
    ! [Xa: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(sup_sup_int(A),B)) ) ).

% less_supI1
tff(fact_5530_less__supI1,axiom,
    ! [Xa: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),A)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(sup_su3973961784419623482d_enat,A),B)) ) ).

% less_supI1
tff(fact_5531_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),ord_max_Code_integer(A,B)) ) ).

% max.strict_coboundedI2
tff(fact_5532_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5533_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5534_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5535_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5536_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5537_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),B)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5538_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),ord_max_Code_integer(A,B)) ) ).

% max.strict_coboundedI1
tff(fact_5539_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5540_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5541_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5542_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5543_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5544_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A)
     => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5545_max_Ostrict__order__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
    <=> ( ( A = ord_max_Code_integer(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5546_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5547_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5548_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5549_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5550_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(ord_max_int(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5551_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5552_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ord_max_Code_integer(B,C)),A)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5553_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5554_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5555_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5556_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5557_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5558_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)
     => ~ ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B),A)
         => ~ aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5559_less__max__iff__disj,axiom,
    ! [Z: code_integer,Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),ord_max_Code_integer(Xa,Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Xa)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5560_less__max__iff__disj,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xa),Y))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xa)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5561_less__max__iff__disj,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(ord_max_rat(Xa),Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xa)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5562_less__max__iff__disj,axiom,
    ! [Z: num,Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xa),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xa)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5563_less__max__iff__disj,axiom,
    ! [Z: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xa)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5564_less__max__iff__disj,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(ord_max_int(Xa),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xa)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5565_less__max__iff__disj,axiom,
    ! [Z: extended_enat,Xa: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xa),Y))
    <=> ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z),Xa)
        | aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5566_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(bot_bot_set_real,Xa) = bot_bot_set_real ).

% boolean_algebra.conj_zero_left
tff(fact_5567_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_o] : inf_inf_set_o(bot_bot_set_o,Xa) = bot_bot_set_o ).

% boolean_algebra.conj_zero_left
tff(fact_5568_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_nat] : inf_inf_set_nat(bot_bot_set_nat,Xa) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_5569_boolean__algebra_Oconj__zero__left,axiom,
    ! [Xa: set_int] : inf_inf_set_int(bot_bot_set_int,Xa) = bot_bot_set_int ).

% boolean_algebra.conj_zero_left
tff(fact_5570_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,bot_bot_set_real) = bot_bot_set_real ).

% boolean_algebra.conj_zero_right
tff(fact_5571_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_o] : inf_inf_set_o(Xa,bot_bot_set_o) = bot_bot_set_o ).

% boolean_algebra.conj_zero_right
tff(fact_5572_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_nat] : inf_inf_set_nat(Xa,bot_bot_set_nat) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_5573_boolean__algebra_Oconj__zero__right,axiom,
    ! [Xa: set_int] : inf_inf_set_int(Xa,bot_bot_set_int) = bot_bot_set_int ).

% boolean_algebra.conj_zero_right
tff(fact_5574_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A4: real,B4: real,C2: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A4),B4)
           => ( aa_real_o(aa_real_fun_real_o(P,B4),C2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),B4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),C2)
                 => aa_real_o(aa_real_fun_real_o(P,A4),C2) ) ) ) )
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
               => ? [D6: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D6)
                    & ! [A4: real,B4: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),X3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B4)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B4),A4)),D6) )
                       => aa_real_o(aa_real_fun_real_o(P,A4),B4) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_5575_set__union,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : set_VEBT_VEBT2(union_VEBT_VEBT(Xs,Ys)) = sup_su6272177626956685416T_VEBT(set_VEBT_VEBT2(Xs),set_VEBT_VEBT2(Ys)) ).

% set_union
tff(fact_5576_set__union,axiom,
    ! [Xs: list_int,Ys: list_int] : set_int2(union_int(Xs,Ys)) = sup_sup_set_int(set_int2(Xs),set_int2(Ys)) ).

% set_union
tff(fact_5577_set__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] : set_nat2(union_nat(Xs,Ys)) = sup_sup_set_nat(set_nat2(Xs),set_nat2(Ys)) ).

% set_union
tff(fact_5578_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( unique6322359934112328802ux_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_5579_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( unique6319869463603278526ux_int(product_Pair_int_int(Q2,R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_5580_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_5581_inf__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] : inf_inf_set_complex(A2,B2) = collect_complex(inf_in8039933884210822705plex_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% inf_set_def
tff(fact_5582_inf__set__def,axiom,
    ! [A2: set_real,B2: set_real] : inf_inf_set_real(A2,B2) = collect_real(inf_inf_fun_real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% inf_set_def
tff(fact_5583_inf__set__def,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : inf_inf_set_list_nat(A2,B2) = collect_list_nat(inf_in1107944134336960767_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% inf_set_def
tff(fact_5584_inf__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : inf_inf_set_set_nat(A2,B2) = collect_set_nat(inf_in7602907407060649817_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% inf_set_def
tff(fact_5585_inf__set__def,axiom,
    ! [A2: set_int,B2: set_int] : inf_inf_set_int(A2,B2) = collect_int(inf_inf_fun_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% inf_set_def
tff(fact_5586_inf__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : inf_inf_set_nat(A2,B2) = collect_nat(inf_inf_fun_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).

% inf_set_def
tff(fact_5587_sup__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] : sup_sup_set_complex(A2,B2) = collect_complex(sup_su37214000559163671plex_o(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).

% sup_set_def
tff(fact_5588_sup__set__def,axiom,
    ! [A2: set_real,B2: set_real] : sup_sup_set_real(A2,B2) = collect_real(sup_sup_fun_real_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).

% sup_set_def
tff(fact_5589_sup__set__def,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] : sup_sup_set_list_nat(A2,B2) = collect_list_nat(sup_su6906368799827444965_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).

% sup_set_def
tff(fact_5590_sup__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] : sup_sup_set_set_nat(A2,B2) = collect_set_nat(sup_su7503774895437956671_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).

% sup_set_def
tff(fact_5591_sup__set__def,axiom,
    ! [A2: set_int,B2: set_int] : sup_sup_set_int(A2,B2) = collect_int(sup_sup_fun_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).

% sup_set_def
tff(fact_5592_sup__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] : sup_sup_set_nat(A2,B2) = collect_nat(sup_sup_fun_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).

% sup_set_def
tff(fact_5593_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_5594_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_real] : sup_sup_set_real(Xa,bot_bot_set_real) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_5595_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_o] : sup_sup_set_o(Xa,bot_bot_set_o) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_5596_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_nat] : sup_sup_set_nat(Xa,bot_bot_set_nat) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_5597_boolean__algebra_Odisj__zero__right,axiom,
    ! [Xa: set_int] : sup_sup_set_int(Xa,bot_bot_set_int) = Xa ).

% boolean_algebra.disj_zero_right
tff(fact_5598_diff__shunt__var,axiom,
    ! [Xa: set_real,Y: set_real] :
      ( ( minus_minus_set_real(Xa,Y) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xa),Y) ) ).

% diff_shunt_var
tff(fact_5599_diff__shunt__var,axiom,
    ! [Xa: set_o,Y: set_o] :
      ( ( minus_minus_set_o(Xa,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(Xa),Y) ) ).

% diff_shunt_var
tff(fact_5600_diff__shunt__var,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xa,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),Y) ) ).

% diff_shunt_var
tff(fact_5601_diff__shunt__var,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(Xa,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) ) ).

% diff_shunt_var
tff(fact_5602_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( eucl_rel_int(aa_int_int(plus_plus_int(A),one_one_int),B,product_Pair_int_int(Q2,R2))
       => eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_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(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5603_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),Nb) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_5604_product__nth,axiom,
    ! [Nb: nat,Xs: list_Code_integer,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys),Nb) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_5605_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_5606_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Nb) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_5607_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Nb) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_5608_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_5609_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys),Nb) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_5610_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys),Nb) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_5611_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys),Nb) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_5612_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys),Nb) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_5613_vebt__buildup_Oelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_5614_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_5615_obtain__set__succ,axiom,
    ! [Xa: nat,Z: nat,A2: set_nat,B2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,B2)
         => ( ( A2 = B2 )
           => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A2),Xa),X_13) ) ) ) ) ).

% obtain_set_succ
tff(fact_5616_obtain__set__pred,axiom,
    ! [Z: nat,Xa: nat,A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xa)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A2),Xa),X_13) ) ) ) ).

% obtain_set_pred
tff(fact_5617_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R2))
       => eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_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(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5618_intind,axiom,
    ! [Ia: nat,Nb: nat,P: fun_int_o,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_int_o(P,Xa)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,Xa)),Ia)) ) ) ).

% intind
tff(fact_5619_intind,axiom,
    ! [Ia: nat,Nb: nat,P: fun_nat_o,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_nat_o(P,Xa)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,Xa)),Ia)) ) ) ).

% intind
tff(fact_5620_intind,axiom,
    ! [Ia: nat,Nb: nat,P: fun_VEBT_VEBT_o,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_VEBT_VEBT_o(P,Xa)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xa),Ia)) ) ) ).

% intind
tff(fact_5621_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_5622_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A),X_13)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),A) ) ) ) ).

% pred_none_empty
tff(fact_5623_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_13)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X) ) ) ) ).

% succ_none_empty
tff(fact_5624_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_5625_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_5626_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_5627_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_5628_replicate__eq__replicate,axiom,
    ! [Ma: nat,Xa: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Ma,Xa) = replicate_VEBT_VEBT(Nb,Y) )
    <=> ( ( Ma = Nb )
        & ( ( Ma != zero_zero_nat )
         => ( Xa = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_5629_length__replicate,axiom,
    ! [Nb: nat,Xa: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xa)) = Nb ).

% length_replicate
tff(fact_5630_length__replicate,axiom,
    ! [Nb: nat,Xa: $o] : size_size_list_o(replicate_o(Nb,(Xa))) = Nb ).

% length_replicate
tff(fact_5631_length__replicate,axiom,
    ! [Nb: nat,Xa: nat] : size_size_list_nat(replicate_nat(Nb,Xa)) = Nb ).

% length_replicate
tff(fact_5632_length__replicate,axiom,
    ! [Nb: nat,Xa: int] : size_size_list_int(replicate_int(Nb,Xa)) = Nb ).

% length_replicate
tff(fact_5633_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_5634_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_5635_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Icc_iff
tff(fact_5636_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(replicate_int(Nb,A)))
         => aa_int_o(P,X4) )
    <=> ( aa_int_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5637_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5638_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5639_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ? [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(replicate_int(Nb,A)))
          & aa_int_o(P,X4) )
    <=> ( aa_int_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5640_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5641_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5642_in__set__replicate,axiom,
    ! [Xa: complex,Nb: nat,Y: complex] :
      ( aa_set_complex_o(member_complex(Xa),set_complex2(replicate_complex(Nb,Y)))
    <=> ( ( Xa = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5643_in__set__replicate,axiom,
    ! [Xa: real,Nb: nat,Y: real] :
      ( aa_set_real_o(member_real(Xa),set_real2(replicate_real(Nb,Y)))
    <=> ( ( Xa = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5644_in__set__replicate,axiom,
    ! [Xa: set_nat,Nb: nat,Y: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(replicate_set_nat(Nb,Y)))
    <=> ( ( Xa = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5645_in__set__replicate,axiom,
    ! [Xa: int,Nb: nat,Y: int] :
      ( aa_set_int_o(member_int(Xa),set_int2(replicate_int(Nb,Y)))
    <=> ( ( Xa = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5646_in__set__replicate,axiom,
    ! [Xa: nat,Nb: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xa),set_nat2(replicate_nat(Nb,Y)))
    <=> ( ( Xa = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5647_in__set__replicate,axiom,
    ! [Xa: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(Xa,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y)))
    <=> ( ( Xa = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5648_nth__replicate,axiom,
    ! [Ia: nat,Nb: nat,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,Xa)),Ia) = Xa ) ) ).

% nth_replicate
tff(fact_5649_nth__replicate,axiom,
    ! [Ia: nat,Nb: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xa)),Ia) = Xa ) ) ).

% nth_replicate
tff(fact_5650_nth__replicate,axiom,
    ! [Ia: nat,Nb: nat,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),Nb)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xa),Ia) = Xa ) ) ).

% nth_replicate
tff(fact_5651_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_5652_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_5653_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_5654_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_5655_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_5656_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_5657_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_5658_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_5659_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_5660_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_5661_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_5662_set__replicate,axiom,
    ! [Nb: nat,Xa: vEBT_VEBT] :
      ( ( Nb != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xa)) = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_5663_set__replicate,axiom,
    ! [Nb: nat,Xa: real] :
      ( ( Nb != zero_zero_nat )
     => ( set_real2(replicate_real(Nb,Xa)) = insert_real(Xa,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_5664_set__replicate,axiom,
    ! [Nb: nat,Xa: $o] :
      ( ( Nb != zero_zero_nat )
     => ( set_o2(replicate_o(Nb,(Xa))) = insert_o((Xa),bot_bot_set_o) ) ) ).

% set_replicate
tff(fact_5665_set__replicate,axiom,
    ! [Nb: nat,Xa: nat] :
      ( ( Nb != zero_zero_nat )
     => ( set_nat2(replicate_nat(Nb,Xa)) = insert_nat(Xa,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_5666_set__replicate,axiom,
    ! [Nb: nat,Xa: int] :
      ( ( Nb != zero_zero_nat )
     => ( set_int2(replicate_int(Nb,Xa)) = insert_int(Xa,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_5667_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Nb) )
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% bounded_nat_set_is_finite
tff(fact_5668_finite__nat__set__iff__bounded,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_5669_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs3: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs3) = A2 ) ).

% finite_list
tff(fact_5670_finite__list,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ? [Xs3: list_nat] : set_nat2(Xs3) = A2 ) ).

% finite_list
tff(fact_5671_finite__list,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ? [Xs3: list_int] : set_int2(Xs3) = A2 ) ).

% finite_list
tff(fact_5672_finite__list,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [Xs3: list_complex] : set_complex2(Xs3) = A2 ) ).

% finite_list
tff(fact_5673_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,Ia: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_do(P),Ia))) ).

% finite_M_bounded_by_nat
tff(fact_5674_finite__lists__length__eq,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_dp(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_5675_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_dq(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_5676_finite__lists__length__eq,axiom,
    ! [A2: set_o,Nb: nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_dr(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_5677_finite__lists__length__eq,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_ds(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_5678_finite__lists__length__eq,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_dt(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_5679_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero_int,product_Pair_int_int(zero_zero_int,K)) ).

% eucl_rel_int_by0
tff(fact_5680_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R2))
     => ( aa_int_int(modulo_modulo_int(K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_5681_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,Xa: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y3: complex] :
            ( aa_set_complex_o(member_complex(Y3),set_complex2(Xs))
           => ( Y3 = Xa ) )
       => ( Xs = replicate_complex(Nb,Xa) ) ) ) ).

% replicate_eqI
tff(fact_5682_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xa: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y3: real] :
            ( aa_set_real_o(member_real(Y3),set_real2(Xs))
           => ( Y3 = Xa ) )
       => ( Xs = replicate_real(Nb,Xa) ) ) ) ).

% replicate_eqI
tff(fact_5683_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Nb: nat,Xa: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Nb )
     => ( ! [Y3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(Y3),set_set_nat2(Xs))
           => ( Y3 = Xa ) )
       => ( Xs = replicate_set_nat(Nb,Xa) ) ) ) ).

% replicate_eqI
tff(fact_5684_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xa: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y3: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y3,set_VEBT_VEBT2(Xs))
           => ( Y3 = Xa ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xa) ) ) ) ).

% replicate_eqI
tff(fact_5685_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,Xa: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y3: $o] :
            ( aa_set_o_o(member_o((Y3)),set_o2(Xs))
           => ( (Y3)
            <=> (Xa) ) )
       => ( Xs = replicate_o(Nb,(Xa)) ) ) ) ).

% replicate_eqI
tff(fact_5686_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xa: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),set_nat2(Xs))
           => ( Y3 = Xa ) )
       => ( Xs = replicate_nat(Nb,Xa) ) ) ) ).

% replicate_eqI
tff(fact_5687_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xa: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y3: int] :
            ( aa_set_int_o(member_int(Y3),set_int2(Xs))
           => ( Y3 = Xa ) )
       => ( Xs = replicate_int(Nb,Xa) ) ) ) ).

% replicate_eqI
tff(fact_5688_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => ( X3 = Xa ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xa) = Xs ) ) ).

% replicate_length_same
tff(fact_5689_replicate__length__same,axiom,
    ! [Xs: list_o,Xa: $o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),set_o2(Xs))
         => ( (X3)
          <=> (Xa) ) )
     => ( replicate_o(size_size_list_o(Xs),(Xa)) = Xs ) ) ).

% replicate_length_same
tff(fact_5690_replicate__length__same,axiom,
    ! [Xs: list_nat,Xa: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => ( X3 = Xa ) )
     => ( replicate_nat(size_size_list_nat(Xs),Xa) = Xs ) ) ).

% replicate_length_same
tff(fact_5691_replicate__length__same,axiom,
    ! [Xs: list_int,Xa: int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => ( X3 = Xa ) )
     => ( replicate_int(size_size_list_int(Xs),Xa) = Xs ) ) ).

% replicate_length_same
tff(fact_5692_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_5693_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_5694_finite__lists__length__le,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_du(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_5695_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_dv(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_5696_finite__lists__length__le,axiom,
    ! [A2: set_o,Nb: nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_dw(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_5697_finite__lists__length__le,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_dx(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_5698_finite__lists__length__le,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_dy(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_5699_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(times_times_int(Q2),L) )
       => eucl_rel_int(K,L,product_Pair_int_int(Q2,zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5700_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,product_Pair_int_int(aa_int_int(divide_divide_int(K),L),aa_int_int(modulo_modulo_int(K),L))) ).

% eucl_rel_int
tff(fact_5701_finite__divisors__nat,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dz,Ma))) ) ).

% finite_divisors_nat
tff(fact_5702_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_5703_set__replicate__Suc,axiom,
    ! [Nb: nat,Xa: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,Nb),Xa)) = insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_5704_set__replicate__Suc,axiom,
    ! [Nb: nat,Xa: real] : set_real2(replicate_real(aa_nat_nat(suc,Nb),Xa)) = insert_real(Xa,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_5705_set__replicate__Suc,axiom,
    ! [Nb: nat,Xa: $o] : set_o2(replicate_o(aa_nat_nat(suc,Nb),(Xa))) = insert_o((Xa),bot_bot_set_o) ).

% set_replicate_Suc
tff(fact_5706_set__replicate__Suc,axiom,
    ! [Nb: nat,Xa: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,Nb),Xa)) = insert_nat(Xa,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_5707_set__replicate__Suc,axiom,
    ! [Nb: nat,Xa: int] : set_int2(replicate_int(aa_nat_nat(suc,Nb),Xa)) = insert_int(Xa,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_5708_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xa: vEBT_VEBT] :
      set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xa)) = $ite(Nb = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT(Xa,bot_bo8194388402131092736T_VEBT)) ).

% set_replicate_conv_if
tff(fact_5709_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xa: real] :
      set_real2(replicate_real(Nb,Xa)) = $ite(Nb = zero_zero_nat,bot_bot_set_real,insert_real(Xa,bot_bot_set_real)) ).

% set_replicate_conv_if
tff(fact_5710_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xa: $o] :
      set_o2(replicate_o(Nb,(Xa))) = $ite(Nb = zero_zero_nat,bot_bot_set_o,insert_o((Xa),bot_bot_set_o)) ).

% set_replicate_conv_if
tff(fact_5711_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xa: nat] :
      set_nat2(replicate_nat(Nb,Xa)) = $ite(Nb = zero_zero_nat,bot_bot_set_nat,insert_nat(Xa,bot_bot_set_nat)) ).

% set_replicate_conv_if
tff(fact_5712_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xa: int] :
      set_int2(replicate_int(Nb,Xa)) = $ite(Nb = zero_zero_nat,bot_bot_set_int,insert_int(Xa,bot_bot_set_int)) ).

% set_replicate_conv_if
tff(fact_5713_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R2))
    <=> ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(L),Q2)),R2) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
              Q2 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5714_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_5715_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ea,K))) ).

% finite_Collect_less_nat
tff(fact_5716_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_real_o(finite_finite_real,collect_real(aTP_Lamm_eb(Nb))) ) ).

% finite_roots_unity
tff(fact_5717_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ec(Nb))) ) ).

% finite_roots_unity
tff(fact_5718_finite__induct__select,axiom,
    ! [S2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [T3: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T3),S2)
             => ( aa_set_complex_o(P,T3)
               => ? [X: complex] :
                    ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(S2,T3))
                    & aa_set_complex_o(P,insert_complex(X,T3)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_5719_finite__induct__select,axiom,
    ! [S2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [T3: set_real] :
              ( aa_set_real_o(ord_less_set_real(T3),S2)
             => ( aa_set_real_o(P,T3)
               => ? [X: real] :
                    ( aa_set_real_o(member_real(X),minus_minus_set_real(S2,T3))
                    & aa_set_real_o(P,insert_real(X,T3)) ) ) )
         => aa_set_real_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_5720_finite__induct__select,axiom,
    ! [S2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [T3: set_o] :
              ( aa_set_o_o(ord_less_set_o(T3),S2)
             => ( aa_set_o_o(P,T3)
               => ? [X: $o] :
                    ( aa_set_o_o(member_o((X)),minus_minus_set_o(S2,T3))
                    & aa_set_o_o(P,insert_o((X),T3)) ) ) )
         => aa_set_o_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_5721_finite__induct__select,axiom,
    ! [S2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T3: set_int] :
              ( aa_set_int_o(ord_less_set_int(T3),S2)
             => ( aa_set_int_o(P,T3)
               => ? [X: int] :
                    ( aa_set_int_o(member_int(X),minus_minus_set_int(S2,T3))
                    & aa_set_int_o(P,insert_int(X,T3)) ) ) )
         => aa_set_int_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_5722_finite__induct__select,axiom,
    ! [S2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T3: set_nat] :
              ( aa_set_nat_o(ord_less_set_nat(T3),S2)
             => ( aa_set_nat_o(P,T3)
               => ? [X: nat] :
                    ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(S2,T3))
                    & aa_set_nat_o(P,insert_nat(X,T3)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_induct_select
tff(fact_5723_remove__induct,axiom,
    ! [P: fun_set_set_nat_o,B2: set_set_nat] :
      ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
     => ( ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
         => aa_set_set_nat_o(P,B2) )
       => ( ! [A7: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A7)
             => ( ( A7 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A7),B2)
                 => ( ! [X: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X),A7)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A7,insert_set_nat(X,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A7) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5724_remove__induct,axiom,
    ! [P: fun_set_complex_o,B2: set_complex] :
      ( aa_set_complex_o(P,bot_bot_set_complex)
     => ( ( ~ aa_set_complex_o(finite3207457112153483333omplex,B2)
         => aa_set_complex_o(P,B2) )
       => ( ! [A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( ( A7 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A7),B2)
                 => ( ! [X: complex] :
                        ( aa_set_complex_o(member_complex(X),A7)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(X,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A7) ) ) ) )
         => aa_set_complex_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5725_remove__induct,axiom,
    ! [P: fun_set_real_o,B2: set_real] :
      ( aa_set_real_o(P,bot_bot_set_real)
     => ( ( ~ aa_set_real_o(finite_finite_real,B2)
         => aa_set_real_o(P,B2) )
       => ( ! [A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ( A7 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A7),B2)
                 => ( ! [X: real] :
                        ( aa_set_real_o(member_real(X),A7)
                       => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(X,bot_bot_set_real))) )
                   => aa_set_real_o(P,A7) ) ) ) )
         => aa_set_real_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5726_remove__induct,axiom,
    ! [P: fun_set_o_o,B2: set_o] :
      ( aa_set_o_o(P,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B2)
         => aa_set_o_o(P,B2) )
       => ( ! [A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ( A7 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A7),B2)
                 => ( ! [X: $o] :
                        ( aa_set_o_o(member_o((X)),A7)
                       => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((X),bot_bot_set_o))) )
                   => aa_set_o_o(P,A7) ) ) ) )
         => aa_set_o_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5727_remove__induct,axiom,
    ! [P: fun_set_int_o,B2: set_int] :
      ( aa_set_int_o(P,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B2)
         => aa_set_int_o(P,B2) )
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A7),B2)
                 => ( ! [X: int] :
                        ( aa_set_int_o(member_int(X),A7)
                       => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(X,bot_bot_set_int))) )
                   => aa_set_int_o(P,A7) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5728_remove__induct,axiom,
    ! [P: fun_set_nat_o,B2: set_nat] :
      ( aa_set_nat_o(P,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B2)
         => aa_set_nat_o(P,B2) )
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A7),B2)
                 => ( ! [X: nat] :
                        ( aa_set_nat_o(member_nat(X),A7)
                       => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(X,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A7) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% remove_induct
tff(fact_5729_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B2)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [A7: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A7)
             => ( ( A7 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A7),B2)
                 => ( ! [X: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X),A7)
                       => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A7,insert_set_nat(X,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P,A7) ) ) ) )
         => aa_set_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5730_finite__remove__induct,axiom,
    ! [B2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( ( A7 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A7),B2)
                 => ( ! [X: complex] :
                        ( aa_set_complex_o(member_complex(X),A7)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(X,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A7) ) ) ) )
         => aa_set_complex_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5731_finite__remove__induct,axiom,
    ! [B2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ( A7 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A7),B2)
                 => ( ! [X: real] :
                        ( aa_set_real_o(member_real(X),A7)
                       => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(X,bot_bot_set_real))) )
                   => aa_set_real_o(P,A7) ) ) ) )
         => aa_set_real_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5732_finite__remove__induct,axiom,
    ! [B2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ( A7 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A7),B2)
                 => ( ! [X: $o] :
                        ( aa_set_o_o(member_o((X)),A7)
                       => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((X),bot_bot_set_o))) )
                   => aa_set_o_o(P,A7) ) ) ) )
         => aa_set_o_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5733_finite__remove__induct,axiom,
    ! [B2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A7),B2)
                 => ( ! [X: int] :
                        ( aa_set_int_o(member_int(X),A7)
                       => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(X,bot_bot_set_int))) )
                   => aa_set_int_o(P,A7) ) ) ) )
         => aa_set_int_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5734_finite__remove__induct,axiom,
    ! [B2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A7),B2)
                 => ( ! [X: nat] :
                        ( aa_set_nat_o(member_nat(X),A7)
                       => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(X,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A7) ) ) ) )
         => aa_set_nat_o(P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5735_set__encode__insert,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Nb),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_5736_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ).

% infinite_remove
tff(fact_5737_infinite__remove,axiom,
    ! [S2: set_real,A: real] :
      ( ~ aa_set_real_o(finite_finite_real,S2)
     => ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real))) ) ).

% infinite_remove
tff(fact_5738_infinite__remove,axiom,
    ! [S2: set_o,A: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S2)
     => ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_5739_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_5740_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_5741_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_5742_finite__maxlen,axiom,
    ! [M6: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M6)
     => ? [N: nat] :
        ! [X: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X)),N) ) ) ).

% finite_maxlen
tff(fact_5743_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( finite_finite_list_o(M6)
     => ? [N: nat] :
        ! [X: list_o] :
          ( member_list_o(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X)),N) ) ) ).

% finite_maxlen
tff(fact_5744_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( finite8100373058378681591st_nat(M6)
     => ? [N: nat] :
        ! [X: list_nat] :
          ( member_list_nat(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X)),N) ) ) ).

% finite_maxlen
tff(fact_5745_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( finite3922522038869484883st_int(M6)
     => ? [N: nat] :
        ! [X: list_int] :
          ( member_list_int(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X)),N) ) ) ).

% finite_maxlen
tff(fact_5746_finite__divisors__int,axiom,
    ! [Ia: int] :
      ( ( Ia != zero_zero_int )
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ed,Ia))) ) ).

% finite_divisors_int
tff(fact_5747_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_5748_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_5749_finite_OemptyI,axiom,
    aa_set_real_o(finite_finite_real,bot_bot_set_real) ).

% finite.emptyI
tff(fact_5750_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_5751_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_5752_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_5753_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_5754_infinite__imp__nonempty,axiom,
    ! [S2: set_real] :
      ( ~ aa_set_real_o(finite_finite_real,S2)
     => ( S2 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_5755_infinite__imp__nonempty,axiom,
    ! [S2: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S2)
     => ( S2 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_5756_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_5757_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_5758_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_5759_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
            & ! [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5760_finite__has__maximal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A2)
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A2)
               => ( aa_o_o(ord_less_eq_o((X3)),(Xa2))
                 => ( (X3)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5761_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A2)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A2)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5762_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A2)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5763_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A2)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5764_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5765_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5766_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
            & ! [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5767_finite__has__minimal,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A2)
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A2)
               => ( aa_o_o(ord_less_eq_o((Xa2)),(X3))
                 => ( (X3)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5768_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A2)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A2)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5769_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A2)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5770_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A2)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5771_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5772_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5773_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A7: set_complex] :
              ( ? [A4: complex] : A = insert_complex(A4,A7)
             => ~ aa_set_complex_o(finite3207457112153483333omplex,A7) ) ) ) ).

% finite.cases
tff(fact_5774_finite_Ocases,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ~ ! [A7: set_real] :
              ( ? [A4: real] : A = insert_real(A4,A7)
             => ~ aa_set_real_o(finite_finite_real,A7) ) ) ) ).

% finite.cases
tff(fact_5775_finite_Ocases,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ~ ! [A7: set_o] :
              ( ? [A4: $o] : A = insert_o((A4),A7)
             => ~ aa_set_o_o(finite_finite_o,A7) ) ) ) ).

% finite.cases
tff(fact_5776_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A4: nat] : A = insert_nat(A4,A7)
             => ~ aa_set_nat_o(finite_finite_nat,A7) ) ) ) ).

% finite.cases
tff(fact_5777_finite_Ocases,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ~ ! [A7: set_int] :
              ( ? [A4: int] : A = insert_int(A4,A7)
             => ~ aa_set_int_o(finite_finite_int,A7) ) ) ) ).

% finite.cases
tff(fact_5778_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
    <=> ( ( A = bot_bot_set_complex )
        | ? [A8: set_complex,A5: complex] :
            ( ( A = insert_complex(A5,A8) )
            & aa_set_complex_o(finite3207457112153483333omplex,A8) ) ) ) ).

% finite.simps
tff(fact_5779_finite_Osimps,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
    <=> ( ( A = bot_bot_set_real )
        | ? [A8: set_real,A5: real] :
            ( ( A = insert_real(A5,A8) )
            & aa_set_real_o(finite_finite_real,A8) ) ) ) ).

% finite.simps
tff(fact_5780_finite_Osimps,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
    <=> ( ( A = bot_bot_set_o )
        | ? [A8: set_o,A5: $o] :
            ( ( A = insert_o((A5),A8) )
            & aa_set_o_o(finite_finite_o,A8) ) ) ) ).

% finite.simps
tff(fact_5781_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
    <=> ( ( A = bot_bot_set_nat )
        | ? [A8: set_nat,A5: nat] :
            ( ( A = insert_nat(A5,A8) )
            & aa_set_nat_o(finite_finite_nat,A8) ) ) ) ).

% finite.simps
tff(fact_5782_finite_Osimps,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
    <=> ( ( A = bot_bot_set_int )
        | ? [A8: set_int,A5: int] :
            ( ( A = insert_int(A5,A8) )
            & aa_set_int_o(finite_finite_int,A8) ) ) ) ).

% finite.simps
tff(fact_5783_finite__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X3: set_nat,F4: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
             => ( ~ aa_set_set_nat_o(member_set_nat(X3),F4)
               => ( aa_set_set_nat_o(P,F4)
                 => aa_set_set_nat_o(P,insert_set_nat(X3,F4)) ) ) )
         => aa_set_set_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5784_finite__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,F4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F4)
             => ( ~ aa_set_complex_o(member_complex(X3),F4)
               => ( aa_set_complex_o(P,F4)
                 => aa_set_complex_o(P,insert_complex(X3,F4)) ) ) )
         => aa_set_complex_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5785_finite__induct,axiom,
    ! [F3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,F4: set_real] :
              ( aa_set_real_o(finite_finite_real,F4)
             => ( ~ aa_set_real_o(member_real(X3),F4)
               => ( aa_set_real_o(P,F4)
                 => aa_set_real_o(P,insert_real(X3,F4)) ) ) )
         => aa_set_real_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5786_finite__induct,axiom,
    ! [F3: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,F4: set_o] :
              ( aa_set_o_o(finite_finite_o,F4)
             => ( ~ aa_set_o_o(member_o((X3)),F4)
               => ( aa_set_o_o(P,F4)
                 => aa_set_o_o(P,insert_o((X3),F4)) ) ) )
         => aa_set_o_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5787_finite__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,F4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F4)
             => ( ~ aa_set_nat_o(member_nat(X3),F4)
               => ( aa_set_nat_o(P,F4)
                 => aa_set_nat_o(P,insert_nat(X3,F4)) ) ) )
         => aa_set_nat_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5788_finite__induct,axiom,
    ! [F3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,F4: set_int] :
              ( aa_set_int_o(finite_finite_int,F4)
             => ( ~ aa_set_int_o(member_int(X3),F4)
               => ( aa_set_int_o(P,F4)
                 => aa_set_int_o(P,insert_int(X3,F4)) ) ) )
         => aa_set_int_o(P,F3) ) ) ) ).

% finite_induct
tff(fact_5789_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X3: set_nat] : aa_set_set_nat_o(P,insert_set_nat(X3,bot_bot_set_set_nat))
         => ( ! [X3: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ aa_set_set_nat_o(member_set_nat(X3),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat(X3,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5790_finite__ne__induct,axiom,
    ! [F3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X3: complex] : aa_set_complex_o(P,insert_complex(X3,bot_bot_set_complex))
         => ( ! [X3: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ aa_set_complex_o(member_complex(X3),F4)
                   => ( aa_set_complex_o(P,F4)
                     => aa_set_complex_o(P,insert_complex(X3,F4)) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5791_finite__ne__induct,axiom,
    ! [F3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X3: real] : aa_set_real_o(P,insert_real(X3,bot_bot_set_real))
         => ( ! [X3: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ aa_set_real_o(member_real(X3),F4)
                   => ( aa_set_real_o(P,F4)
                     => aa_set_real_o(P,insert_real(X3,F4)) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5792_finite__ne__induct,axiom,
    ! [F3: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( ( F3 != bot_bot_set_o )
       => ( ! [X3: $o] : aa_set_o_o(P,insert_o((X3),bot_bot_set_o))
         => ( ! [X3: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( ( F4 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o((X3)),F4)
                   => ( aa_set_o_o(P,F4)
                     => aa_set_o_o(P,insert_o((X3),F4)) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5793_finite__ne__induct,axiom,
    ! [F3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X3: nat] : aa_set_nat_o(P,insert_nat(X3,bot_bot_set_nat))
         => ( ! [X3: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat(X3),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,insert_nat(X3,F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5794_finite__ne__induct,axiom,
    ! [F3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X3: int] : aa_set_int_o(P,insert_int(X3,bot_bot_set_int))
         => ( ! [X3: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int(X3),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int(X3,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5795_infinite__finite__induct,axiom,
    ! [P: fun_set_set_nat_o,A2: set_set_nat] :
      ( ! [A7: set_set_nat] :
          ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A7)
         => aa_set_set_nat_o(P,A7) )
     => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
       => ( ! [X3: set_nat,F4: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
             => ( ~ aa_set_set_nat_o(member_set_nat(X3),F4)
               => ( aa_set_set_nat_o(P,F4)
                 => aa_set_set_nat_o(P,insert_set_nat(X3,F4)) ) ) )
         => aa_set_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5796_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_o,A2: set_complex] :
      ( ! [A7: set_complex] :
          ( ~ aa_set_complex_o(finite3207457112153483333omplex,A7)
         => aa_set_complex_o(P,A7) )
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,F4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F4)
             => ( ~ aa_set_complex_o(member_complex(X3),F4)
               => ( aa_set_complex_o(P,F4)
                 => aa_set_complex_o(P,insert_complex(X3,F4)) ) ) )
         => aa_set_complex_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5797_infinite__finite__induct,axiom,
    ! [P: fun_set_real_o,A2: set_real] :
      ( ! [A7: set_real] :
          ( ~ aa_set_real_o(finite_finite_real,A7)
         => aa_set_real_o(P,A7) )
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,F4: set_real] :
              ( aa_set_real_o(finite_finite_real,F4)
             => ( ~ aa_set_real_o(member_real(X3),F4)
               => ( aa_set_real_o(P,F4)
                 => aa_set_real_o(P,insert_real(X3,F4)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5798_infinite__finite__induct,axiom,
    ! [P: fun_set_o_o,A2: set_o] :
      ( ! [A7: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A7)
         => aa_set_o_o(P,A7) )
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,F4: set_o] :
              ( aa_set_o_o(finite_finite_o,F4)
             => ( ~ aa_set_o_o(member_o((X3)),F4)
               => ( aa_set_o_o(P,F4)
                 => aa_set_o_o(P,insert_o((X3),F4)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5799_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_o,A2: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A7)
         => aa_set_nat_o(P,A7) )
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,F4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F4)
             => ( ~ aa_set_nat_o(member_nat(X3),F4)
               => ( aa_set_nat_o(P,F4)
                 => aa_set_nat_o(P,insert_nat(X3,F4)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5800_infinite__finite__induct,axiom,
    ! [P: fun_set_int_o,A2: set_int] :
      ( ! [A7: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A7)
         => aa_set_int_o(P,A7) )
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,F4: set_int] :
              ( aa_set_int_o(finite_finite_int,F4)
             => ( ~ aa_set_int_o(member_int(X3),F4)
               => ( aa_set_int_o(P,F4)
                 => aa_set_int_o(P,insert_int(X3,F4)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5801_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A2)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A4: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( aa_set_set_nat_o(member_set_nat(A4),A2)
                 => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F4),A2)
                   => ( ~ aa_set_set_nat_o(member_set_nat(A4),F4)
                     => ( aa_set_set_nat_o(P,F4)
                       => aa_set_set_nat_o(P,insert_set_nat(A4,F4)) ) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5802_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A2)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A4: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( aa_set_complex_o(member_complex(A4),A2)
                 => ( aa_set_complex_o(ord_le211207098394363844omplex(F4),A2)
                   => ( ~ aa_set_complex_o(member_complex(A4),F4)
                     => ( aa_set_complex_o(P,F4)
                       => aa_set_complex_o(P,insert_complex(A4,F4)) ) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5803_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(ord_less_eq_set_real(F3),A2)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A4: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( aa_set_real_o(member_real(A4),A2)
                 => ( aa_set_real_o(ord_less_eq_set_real(F4),A2)
                   => ( ~ aa_set_real_o(member_real(A4),F4)
                     => ( aa_set_real_o(P,F4)
                       => aa_set_real_o(P,insert_real(A4,F4)) ) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5804_finite__subset__induct_H,axiom,
    ! [F3: set_o,A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(ord_less_eq_set_o(F3),A2)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A4: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( aa_set_o_o(member_o((A4)),A2)
                 => ( aa_set_o_o(ord_less_eq_set_o(F4),A2)
                   => ( ~ aa_set_o_o(member_o((A4)),F4)
                     => ( aa_set_o_o(P,F4)
                       => aa_set_o_o(P,insert_o((A4),F4)) ) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5805_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(ord_less_eq_set_int(F3),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A4: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( aa_set_int_o(member_int(A4),A2)
                 => ( aa_set_int_o(ord_less_eq_set_int(F4),A2)
                   => ( ~ aa_set_int_o(member_int(A4),F4)
                     => ( aa_set_int_o(P,F4)
                       => aa_set_int_o(P,insert_int(A4,F4)) ) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5806_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F3),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A4: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( aa_set_nat_o(member_nat(A4),A2)
                 => ( aa_set_nat_o(ord_less_eq_set_nat(F4),A2)
                   => ( ~ aa_set_nat_o(member_nat(A4),F4)
                     => ( aa_set_nat_o(P,F4)
                       => aa_set_nat_o(P,insert_nat(A4,F4)) ) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5807_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A2)
       => ( aa_set_set_nat_o(P,bot_bot_set_set_nat)
         => ( ! [A4: set_nat,F4: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
               => ( aa_set_set_nat_o(member_set_nat(A4),A2)
                 => ( ~ aa_set_set_nat_o(member_set_nat(A4),F4)
                   => ( aa_set_set_nat_o(P,F4)
                     => aa_set_set_nat_o(P,insert_set_nat(A4,F4)) ) ) ) )
           => aa_set_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5808_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A2)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A4: complex,F4: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F4)
               => ( aa_set_complex_o(member_complex(A4),A2)
                 => ( ~ aa_set_complex_o(member_complex(A4),F4)
                   => ( aa_set_complex_o(P,F4)
                     => aa_set_complex_o(P,insert_complex(A4,F4)) ) ) ) )
           => aa_set_complex_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5809_finite__subset__induct,axiom,
    ! [F3: set_real,A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F3)
     => ( aa_set_real_o(ord_less_eq_set_real(F3),A2)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A4: real,F4: set_real] :
                ( aa_set_real_o(finite_finite_real,F4)
               => ( aa_set_real_o(member_real(A4),A2)
                 => ( ~ aa_set_real_o(member_real(A4),F4)
                   => ( aa_set_real_o(P,F4)
                     => aa_set_real_o(P,insert_real(A4,F4)) ) ) ) )
           => aa_set_real_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5810_finite__subset__induct,axiom,
    ! [F3: set_o,A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F3)
     => ( aa_set_o_o(ord_less_eq_set_o(F3),A2)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A4: $o,F4: set_o] :
                ( aa_set_o_o(finite_finite_o,F4)
               => ( aa_set_o_o(member_o((A4)),A2)
                 => ( ~ aa_set_o_o(member_o((A4)),F4)
                   => ( aa_set_o_o(P,F4)
                     => aa_set_o_o(P,insert_o((A4),F4)) ) ) ) )
           => aa_set_o_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5811_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( aa_set_int_o(ord_less_eq_set_int(F3),A2)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A4: int,F4: set_int] :
                ( aa_set_int_o(finite_finite_int,F4)
               => ( aa_set_int_o(member_int(A4),A2)
                 => ( ~ aa_set_int_o(member_int(A4),F4)
                   => ( aa_set_int_o(P,F4)
                     => aa_set_int_o(P,insert_int(A4,F4)) ) ) ) )
           => aa_set_int_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5812_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F3),A2)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A4: nat,F4: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F4)
               => ( aa_set_nat_o(member_nat(A4),A2)
                 => ( ~ aa_set_nat_o(member_nat(A4),F4)
                   => ( aa_set_nat_o(P,F4)
                     => aa_set_nat_o(P,insert_nat(A4,F4)) ) ) ) )
           => aa_set_nat_o(P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5813_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A2)
     => ( aa_set_set_nat_o(P,A2)
       => ( ! [A4: set_nat,A7: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A7)
             => ( aa_set_set_nat_o(member_set_nat(A4),A7)
               => ( aa_set_set_nat_o(P,A7)
                 => aa_set_set_nat_o(P,minus_2163939370556025621et_nat(A7,insert_set_nat(A4,bot_bot_set_set_nat))) ) ) )
         => aa_set_set_nat_o(P,bot_bot_set_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_5814_finite__empty__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(P,A2)
       => ( ! [A4: complex,A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( aa_set_complex_o(member_complex(A4),A7)
               => ( aa_set_complex_o(P,A7)
                 => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(A4,bot_bot_set_complex))) ) ) )
         => aa_set_complex_o(P,bot_bot_set_complex) ) ) ) ).

% finite_empty_induct
tff(fact_5815_finite__empty__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,A2)
       => ( ! [A4: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( aa_set_real_o(member_real(A4),A7)
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(A4,bot_bot_set_real))) ) ) )
         => aa_set_real_o(P,bot_bot_set_real) ) ) ) ).

% finite_empty_induct
tff(fact_5816_finite__empty__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,A2)
       => ( ! [A4: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( aa_set_o_o(member_o((A4)),A7)
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((A4),bot_bot_set_o))) ) ) )
         => aa_set_o_o(P,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_5817_finite__empty__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,A2)
       => ( ! [A4: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( aa_set_int_o(member_int(A4),A7)
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(A4,bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_5818_finite__empty__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,A2)
       => ( ! [A4: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( aa_set_nat_o(member_nat(A4),A7)
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(A4,bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_5819_infinite__coinduct,axiom,
    ! [X6: fun_set_complex_o,A2: set_complex] :
      ( aa_set_complex_o(X6,A2)
     => ( ! [A7: set_complex] :
            ( aa_set_complex_o(X6,A7)
           => ? [X: complex] :
                ( aa_set_complex_o(member_complex(X),A7)
                & ( aa_set_complex_o(X6,minus_811609699411566653omplex(A7,insert_complex(X,bot_bot_set_complex)))
                  | ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A7,insert_complex(X,bot_bot_set_complex))) ) ) )
       => ~ aa_set_complex_o(finite3207457112153483333omplex,A2) ) ) ).

% infinite_coinduct
tff(fact_5820_infinite__coinduct,axiom,
    ! [X6: fun_set_real_o,A2: set_real] :
      ( aa_set_real_o(X6,A2)
     => ( ! [A7: set_real] :
            ( aa_set_real_o(X6,A7)
           => ? [X: real] :
                ( aa_set_real_o(member_real(X),A7)
                & ( aa_set_real_o(X6,minus_minus_set_real(A7,insert_real(X,bot_bot_set_real)))
                  | ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(A7,insert_real(X,bot_bot_set_real))) ) ) )
       => ~ aa_set_real_o(finite_finite_real,A2) ) ) ).

% infinite_coinduct
tff(fact_5821_infinite__coinduct,axiom,
    ! [X6: fun_set_o_o,A2: set_o] :
      ( aa_set_o_o(X6,A2)
     => ( ! [A7: set_o] :
            ( aa_set_o_o(X6,A7)
           => ? [X: $o] :
                ( aa_set_o_o(member_o((X)),A7)
                & ( aa_set_o_o(X6,minus_minus_set_o(A7,insert_o((X),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A7,insert_o((X),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A2) ) ) ).

% infinite_coinduct
tff(fact_5822_infinite__coinduct,axiom,
    ! [X6: fun_set_int_o,A2: set_int] :
      ( aa_set_int_o(X6,A2)
     => ( ! [A7: set_int] :
            ( aa_set_int_o(X6,A7)
           => ? [X: int] :
                ( aa_set_int_o(member_int(X),A7)
                & ( aa_set_int_o(X6,minus_minus_set_int(A7,insert_int(X,bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A7,insert_int(X,bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A2) ) ) ).

% infinite_coinduct
tff(fact_5823_infinite__coinduct,axiom,
    ! [X6: fun_set_nat_o,A2: set_nat] :
      ( aa_set_nat_o(X6,A2)
     => ( ! [A7: set_nat] :
            ( aa_set_nat_o(X6,A7)
           => ? [X: nat] :
                ( aa_set_nat_o(member_nat(X),A7)
                & ( aa_set_nat_o(X6,minus_minus_set_nat(A7,insert_nat(X,bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A7,insert_nat(X,bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% infinite_coinduct
tff(fact_5824_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_5825_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ee(Nb),C))) ) ).

% finite_nth_roots
tff(fact_5826_finite__linorder__min__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B4: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ! [X: $o] :
                    ( aa_set_o_o(member_o((X)),A7)
                   => aa_o_o(ord_less_o((B4)),(X)) )
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,insert_o((B4),A7)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5827_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B4: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X: real] :
                    ( aa_set_real_o(member_real(X),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B4),X) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B4,A7)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5828_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B4: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X: rat] :
                    ( aa_set_rat_o(member_rat(X),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),X) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B4,A7)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5829_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B4: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X: num] :
                    ( aa_set_num_o(member_num(X),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B4),X) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B4,A7)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5830_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B4: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X: nat] :
                    ( aa_set_nat_o(member_nat(X),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B4),X) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B4,A7)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5831_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B4: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X: int] :
                    ( aa_set_int_o(member_int(X),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B4),X) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B4,A7)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5832_finite__linorder__min__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [B4: extended_enat,A7: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A7)
             => ( ! [X: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A7)
                   => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,B4),X) )
               => ( aa_set3337841544853333262enat_o(P,A7)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(B4,A7)) ) ) )
         => aa_set3337841544853333262enat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5833_finite__linorder__max__induct,axiom,
    ! [A2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B4: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ! [X: $o] :
                    ( aa_set_o_o(member_o((X)),A7)
                   => aa_o_o(ord_less_o((X)),(B4)) )
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,insert_o((B4),A7)) ) ) )
         => aa_set_o_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5834_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B4: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X: real] :
                    ( aa_set_real_o(member_real(X),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X),B4) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B4,A7)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5835_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B4: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X: rat] :
                    ( aa_set_rat_o(member_rat(X),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),B4) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B4,A7)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5836_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B4: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X: num] :
                    ( aa_set_num_o(member_num(X),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X),B4) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B4,A7)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5837_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B4: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X: nat] :
                    ( aa_set_nat_o(member_nat(X),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),B4) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B4,A7)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5838_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B4: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X: int] :
                    ( aa_set_int_o(member_int(X),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X),B4) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B4,A7)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5839_finite__linorder__max__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se8123728892507317397enat_o] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A2)
     => ( aa_set3337841544853333262enat_o(P,bot_bo7653980558646680370d_enat)
       => ( ! [B4: extended_enat,A7: set_Extended_enat] :
              ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,A7)
             => ( ! [X: extended_enat] :
                    ( aa_set3337841544853333262enat_o(member_Extended_enat(X),A7)
                   => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X),B4) )
               => ( aa_set3337841544853333262enat_o(P,A7)
                 => aa_set3337841544853333262enat_o(P,insert_Extended_enat(B4,A7)) ) ) )
         => aa_set3337841544853333262enat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5840_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,S3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S3)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X3)) )
               => ( aa_set_complex_o(P,S3)
                 => aa_set_complex_o(P,insert_complex(X3,S3)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5841_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,S3: set_real] :
              ( aa_set_real_o(finite_finite_real,S3)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X3)) )
               => ( aa_set_real_o(P,S3)
                 => aa_set_real_o(P,insert_real(X3,S3)) ) ) )
         => aa_set_real_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5842_finite__ranking__induct,axiom,
    ! [S2: set_o,P: fun_set_o_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,S3: set_o] :
              ( aa_set_o_o(finite_finite_o,S3)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(Y4))),aa_o_rat(F,(X3))) )
               => ( aa_set_o_o(P,S3)
                 => aa_set_o_o(P,insert_o((X3),S3)) ) ) )
         => aa_set_o_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5843_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,S3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S3)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X3)) )
               => ( aa_set_nat_o(P,S3)
                 => aa_set_nat_o(P,insert_nat(X3,S3)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5844_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,S3: set_int] :
              ( aa_set_int_o(finite_finite_int,S3)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S3)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X3)) )
               => ( aa_set_int_o(P,S3)
                 => aa_set_int_o(P,insert_int(X3,S3)) ) ) )
         => aa_set_int_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5845_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: fun_set_complex_o,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,S3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S3)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X3)) )
               => ( aa_set_complex_o(P,S3)
                 => aa_set_complex_o(P,insert_complex(X3,S3)) ) ) )
         => aa_set_complex_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5846_finite__ranking__induct,axiom,
    ! [S2: set_real,P: fun_set_real_o,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,S3: set_real] :
              ( aa_set_real_o(finite_finite_real,S3)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,Y4)),aa_real_num(F,X3)) )
               => ( aa_set_real_o(P,S3)
                 => aa_set_real_o(P,insert_real(X3,S3)) ) ) )
         => aa_set_real_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5847_finite__ranking__induct,axiom,
    ! [S2: set_o,P: fun_set_o_o,F: fun_o_num] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,S3: set_o] :
              ( aa_set_o_o(finite_finite_o,S3)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_o_num(F,(Y4))),aa_o_num(F,(X3))) )
               => ( aa_set_o_o(P,S3)
                 => aa_set_o_o(P,insert_o((X3),S3)) ) ) )
         => aa_set_o_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5848_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: fun_set_nat_o,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,S3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S3)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Y4)),aa_nat_num(F,X3)) )
               => ( aa_set_nat_o(P,S3)
                 => aa_set_nat_o(P,insert_nat(X3,S3)) ) ) )
         => aa_set_nat_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5849_finite__ranking__induct,axiom,
    ! [S2: set_int,P: fun_set_int_o,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,S3: set_int] :
              ( aa_set_int_o(finite_finite_int,S3)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S3)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,Y4)),aa_int_num(F,X3)) )
               => ( aa_set_int_o(P,S3)
                 => aa_set_int_o(P,insert_int(X3,S3)) ) ) )
         => aa_set_int_o(P,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_5850_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5851_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_5852_ex__min__if__finite,axiom,
    ! [S2: set_o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ( S2 != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),S2)
            & ~ ? [Xa2: $o] :
                  ( aa_set_o_o(member_o((Xa2)),S2)
                  & aa_o_o(ord_less_o((Xa2)),(X3)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5853_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( S2 != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),S2)
            & ~ ? [Xa2: real] :
                  ( aa_set_real_o(member_real(Xa2),S2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5854_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S2)
     => ( ( S2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),S2)
            & ~ ? [Xa2: rat] :
                  ( aa_set_rat_o(member_rat(Xa2),S2)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5855_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( aa_set_num_o(finite_finite_num,S2)
     => ( ( S2 != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),S2)
            & ~ ? [Xa2: num] :
                  ( aa_set_num_o(member_num(Xa2),S2)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5856_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),S2)
            & ~ ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),S2)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5857_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),S2)
            & ~ ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),S2)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5858_ex__min__if__finite,axiom,
    ! [S2: set_Extended_enat] :
      ( aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,S2)
     => ( ( S2 != bot_bo7653980558646680370d_enat )
       => ? [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),S2)
            & ~ ? [Xa2: extended_enat] :
                  ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),S2)
                  & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5859_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),X6)
           => ? [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),X6)
                & aa_o_o(ord_less_o((X3)),(Xa2)) ) )
       => ~ aa_set_o_o(finite_finite_o,X6) ) ) ).

% infinite_growing
tff(fact_5860_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),X6)
           => ? [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),X6)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Xa2) ) )
       => ~ aa_set_real_o(finite_finite_real,X6) ) ) ).

% infinite_growing
tff(fact_5861_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),X6)
           => ? [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),X6)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Xa2) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X6) ) ) ).

% infinite_growing
tff(fact_5862_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( aa_set_num_o(member_num(X3),X6)
           => ? [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),X6)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Xa2) ) )
       => ~ aa_set_num_o(finite_finite_num,X6) ) ) ).

% infinite_growing
tff(fact_5863_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X6)
           => ? [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),X6)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa2) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X6) ) ) ).

% infinite_growing
tff(fact_5864_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X6)
           => ? [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),X6)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Xa2) ) )
       => ~ aa_set_int_o(finite_finite_int,X6) ) ) ).

% infinite_growing
tff(fact_5865_infinite__growing,axiom,
    ! [X6: set_Extended_enat] :
      ( ( X6 != bot_bo7653980558646680370d_enat )
     => ( ! [X3: extended_enat] :
            ( aa_set3337841544853333262enat_o(member_Extended_enat(X3),X6)
           => ? [Xa2: extended_enat] :
                ( aa_set3337841544853333262enat_o(member_Extended_enat(Xa2),X6)
                & aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,X3),Xa2) ) )
       => ~ aa_set3337841544853333262enat_o(finite4001608067531595151d_enat,X6) ) ) ).

% infinite_growing
tff(fact_5866_artanh__def,axiom,
    ! [Xa: real] : aa_real_real(artanh_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5867_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_complex,Y: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ef(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ef(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_eg(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5868_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_complex,Y: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_eh(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_eh(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_ei(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5869_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_complex,Y: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_ej(I5),Xa)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_ej(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ek(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5870_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xa: fun_complex_complex,Y: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_el(I5),Xa)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_el(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_em(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5871_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_en(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_en(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_eo(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5872_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ep(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ep(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_eq(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5873_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_er(I5),Xa)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_er(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_es(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5874_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xa: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_et(I5),Xa)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_et(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_eu(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5875_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ev(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ev(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ew(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5876_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ex(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ex(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ey(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5877_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_complex,Y: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ez(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ez(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_fa(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5878_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_complex,Y: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_fb(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_fb(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_fc(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5879_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_complex,Y: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_fd(I5),Xa)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_fd(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_fe(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5880_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xa: fun_complex_complex,Y: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_ff(I5),Xa)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_ff(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_fg(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5881_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_fh(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_fh(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_fi(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5882_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_fj(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_fj(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_fk(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5883_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_fl(I5),Xa)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_fl(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_fm(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5884_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xa: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_fn(I5),Xa)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_fn(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_fo(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5885_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fp(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fp(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_fq(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5886_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_fr(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_fr(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_fs(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5887_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(product_Pair_int_int(Q2,R2)) = aa_int_int(plus_plus_int(Q2),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_5888_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5889_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5890_vebt__buildup_Opelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5891_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_nat_nat,X2: nat] : size_option_nat(Xa,some_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Xa,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5892_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(Xa,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xa,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5893_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_num_nat,X2: num] : size_option_num(Xa,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xa,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5894_verit__minus__simplify_I4_J,axiom,
    ! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).

% verit_minus_simplify(4)
tff(fact_5895_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_5896_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B)) = B ).

% verit_minus_simplify(4)
tff(fact_5897_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_5898_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ).

% verit_minus_simplify(4)
tff(fact_5899_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_5900_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_5901_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_5902_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_5903_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_5904_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_5905_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_5906_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_5907_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_5908_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_5909_Compl__anti__mono,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(B2)),uminus5710092332889474511et_nat(A2)) ) ).

% Compl_anti_mono
tff(fact_5910_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(A2)),uminus5710092332889474511et_nat(B2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ).

% Compl_subset_Compl_iff
tff(fact_5911_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_5912_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_5913_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_5914_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_5915_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_5916_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_5917_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_5918_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_5919_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_5920_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_5921_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_5922_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_5923_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_5924_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_5925_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_5926_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_5927_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_5928_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_5929_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_5930_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_5931_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_5932_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_5933_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_5934_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_5935_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_5936_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_5937_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_5938_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_5939_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_5940_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_5941_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_5942_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5943_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5944_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5945_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5946_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5947_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_5948_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A),B)) ).

% mult_minus_left
tff(fact_5949_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_5950_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_5951_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_5952_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_5953_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(times_times_int(A),B) ).

% minus_mult_minus
tff(fact_5954_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_5955_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_5956_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_5957_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_5958_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A),B)) ).

% mult_minus_right
tff(fact_5959_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_5960_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_5961_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_5962_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_5963_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_5964_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_5965_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).

% add_minus_cancel
tff(fact_5966_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_5967_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).

% minus_add_cancel
tff(fact_5968_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(plus_plus_int(A),B)) = B ).

% minus_add_cancel
tff(fact_5969_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_5970_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_5971_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).

% minus_add_cancel
tff(fact_5972_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_5973_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_5974_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_5975_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_5976_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_5977_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% minus_diff_eq
tff(fact_5978_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A),B)) = aa_int_int(minus_minus_int(B),A) ).

% minus_diff_eq
tff(fact_5979_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_5980_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_5981_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_5982_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(divide_divide_int(A),B) ).

% div_minus_minus
tff(fact_5983_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_5984_ln__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_5985_ln__inj__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( ( aa_real_real(ln_ln_real,Xa) = aa_real_real(ln_ln_real,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_5986_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B)) ).

% mod_minus_minus
tff(fact_5987_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_5988_Compl__disjoint,axiom,
    ! [A2: set_real] : inf_inf_set_real(A2,uminus612125837232591019t_real(A2)) = bot_bot_set_real ).

% Compl_disjoint
tff(fact_5989_Compl__disjoint,axiom,
    ! [A2: set_o] : inf_inf_set_o(A2,uminus_uminus_set_o(A2)) = bot_bot_set_o ).

% Compl_disjoint
tff(fact_5990_Compl__disjoint,axiom,
    ! [A2: set_nat] : inf_inf_set_nat(A2,uminus5710092332889474511et_nat(A2)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_5991_Compl__disjoint,axiom,
    ! [A2: set_int] : inf_inf_set_int(A2,uminus1532241313380277803et_int(A2)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_5992_Compl__disjoint2,axiom,
    ! [A2: set_real] : inf_inf_set_real(uminus612125837232591019t_real(A2),A2) = bot_bot_set_real ).

% Compl_disjoint2
tff(fact_5993_Compl__disjoint2,axiom,
    ! [A2: set_o] : inf_inf_set_o(uminus_uminus_set_o(A2),A2) = bot_bot_set_o ).

% Compl_disjoint2
tff(fact_5994_Compl__disjoint2,axiom,
    ! [A2: set_nat] : inf_inf_set_nat(uminus5710092332889474511et_nat(A2),A2) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_5995_Compl__disjoint2,axiom,
    ! [A2: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(A2),A2) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_5996_real__add__minus__iff,axiom,
    ! [Xa: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xa = A ) ) ).

% real_add_minus_iff
tff(fact_5997_Diff__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,uminus5710092332889474511et_nat(B2)) = inf_inf_set_nat(A2,B2) ).

% Diff_Compl
tff(fact_5998_Compl__Diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(minus_minus_set_nat(A2,B2)) = sup_sup_set_nat(uminus5710092332889474511et_nat(A2),B2) ).

% Compl_Diff_eq
tff(fact_5999_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_6000_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_6001_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_6002_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_6003_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_6004_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_6005_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_6006_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_6007_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_6008_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_6009_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_6010_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_6011_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_6012_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_6013_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_6014_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_6015_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_6016_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_6017_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_6018_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_6019_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_6020_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_6021_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_6022_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_6023_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_6024_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_6025_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_6026_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_6027_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_6028_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_6029_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_6030_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_6031_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_6032_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_6033_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_6034_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_6035_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_6036_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_6037_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_6038_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_6039_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_6040_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_6041_diff__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_6042_diff__0,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_6043_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_6044_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_6045_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_6046_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_6047_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(minus_minus_int(zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_6048_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_6049_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_6050_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_6051_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_6052_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_6053_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Ma)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_6054_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_6055_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_6056_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_6057_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_6058_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_6059_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_6060_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_6061_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_6062_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_6063_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_6064_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_6065_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_6066_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% uminus_add_conv_diff
tff(fact_6067_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(minus_minus_int(B),A) ).

% uminus_add_conv_diff
tff(fact_6068_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_6069_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_6070_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_6071_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_6072_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(plus_plus_int(A),B) ).

% diff_minus_eq_add
tff(fact_6073_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_6074_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_6075_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).

% diff_minus_eq_add
tff(fact_6076_divide__minus1,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xa) ).

% divide_minus1
tff(fact_6077_divide__minus1,axiom,
    ! [Xa: complex] : aa_complex_complex(divide1717551699836669952omplex(Xa),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xa) ).

% divide_minus1
tff(fact_6078_divide__minus1,axiom,
    ! [Xa: rat] : aa_rat_rat(divide_divide_rat(Xa),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xa) ).

% divide_minus1
tff(fact_6079_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_6080_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_6081_inf__compl__bot__left1,axiom,
    ! [Xa: set_real,Y: set_real] : inf_inf_set_real(uminus612125837232591019t_real(Xa),inf_inf_set_real(Xa,Y)) = bot_bot_set_real ).

% inf_compl_bot_left1
tff(fact_6082_inf__compl__bot__left1,axiom,
    ! [Xa: set_o,Y: set_o] : inf_inf_set_o(uminus_uminus_set_o(Xa),inf_inf_set_o(Xa,Y)) = bot_bot_set_o ).

% inf_compl_bot_left1
tff(fact_6083_inf__compl__bot__left1,axiom,
    ! [Xa: set_nat,Y: set_nat] : inf_inf_set_nat(uminus5710092332889474511et_nat(Xa),inf_inf_set_nat(Xa,Y)) = bot_bot_set_nat ).

% inf_compl_bot_left1
tff(fact_6084_inf__compl__bot__left1,axiom,
    ! [Xa: set_int,Y: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(Xa),inf_inf_set_int(Xa,Y)) = bot_bot_set_int ).

% inf_compl_bot_left1
tff(fact_6085_inf__compl__bot__left2,axiom,
    ! [Xa: set_real,Y: set_real] : inf_inf_set_real(Xa,inf_inf_set_real(uminus612125837232591019t_real(Xa),Y)) = bot_bot_set_real ).

% inf_compl_bot_left2
tff(fact_6086_inf__compl__bot__left2,axiom,
    ! [Xa: set_o,Y: set_o] : inf_inf_set_o(Xa,inf_inf_set_o(uminus_uminus_set_o(Xa),Y)) = bot_bot_set_o ).

% inf_compl_bot_left2
tff(fact_6087_inf__compl__bot__left2,axiom,
    ! [Xa: set_nat,Y: set_nat] : inf_inf_set_nat(Xa,inf_inf_set_nat(uminus5710092332889474511et_nat(Xa),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left2
tff(fact_6088_inf__compl__bot__left2,axiom,
    ! [Xa: set_int,Y: set_int] : inf_inf_set_int(Xa,inf_inf_set_int(uminus1532241313380277803et_int(Xa),Y)) = bot_bot_set_int ).

% inf_compl_bot_left2
tff(fact_6089_inf__compl__bot__right,axiom,
    ! [Xa: set_real,Y: set_real] : inf_inf_set_real(Xa,inf_inf_set_real(Y,uminus612125837232591019t_real(Xa))) = bot_bot_set_real ).

% inf_compl_bot_right
tff(fact_6090_inf__compl__bot__right,axiom,
    ! [Xa: set_o,Y: set_o] : inf_inf_set_o(Xa,inf_inf_set_o(Y,uminus_uminus_set_o(Xa))) = bot_bot_set_o ).

% inf_compl_bot_right
tff(fact_6091_inf__compl__bot__right,axiom,
    ! [Xa: set_nat,Y: set_nat] : inf_inf_set_nat(Xa,inf_inf_set_nat(Y,uminus5710092332889474511et_nat(Xa))) = bot_bot_set_nat ).

% inf_compl_bot_right
tff(fact_6092_inf__compl__bot__right,axiom,
    ! [Xa: set_int,Y: set_int] : inf_inf_set_int(Xa,inf_inf_set_int(Y,uminus1532241313380277803et_int(Xa))) = bot_bot_set_int ).

% inf_compl_bot_right
tff(fact_6093_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_real] : inf_inf_set_real(uminus612125837232591019t_real(Xa),Xa) = bot_bot_set_real ).

% boolean_algebra.conj_cancel_left
tff(fact_6094_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_o] : inf_inf_set_o(uminus_uminus_set_o(Xa),Xa) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_left
tff(fact_6095_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_nat] : inf_inf_set_nat(uminus5710092332889474511et_nat(Xa),Xa) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_6096_boolean__algebra_Oconj__cancel__left,axiom,
    ! [Xa: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(Xa),Xa) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_left
tff(fact_6097_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_real] : inf_inf_set_real(Xa,uminus612125837232591019t_real(Xa)) = bot_bot_set_real ).

% boolean_algebra.conj_cancel_right
tff(fact_6098_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_o] : inf_inf_set_o(Xa,uminus_uminus_set_o(Xa)) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_right
tff(fact_6099_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_nat] : inf_inf_set_nat(Xa,uminus5710092332889474511et_nat(Xa)) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_6100_boolean__algebra_Oconj__cancel__right,axiom,
    ! [Xa: set_int] : inf_inf_set_int(Xa,uminus1532241313380277803et_int(Xa)) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_right
tff(fact_6101_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(B),A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_6102_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_mod_self1
tff(fact_6103_subset__Compl__singleton,axiom,
    ! [A2: set_complex,B: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A2),uminus8566677241136511917omplex(insert_complex(B,bot_bot_set_complex)))
    <=> ~ aa_set_complex_o(member_complex(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6104_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A2),uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat)))
    <=> ~ aa_set_set_nat_o(member_set_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6105_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(insert_real(B,bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6106_subset__Compl__singleton,axiom,
    ! [A2: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),uminus_uminus_set_o(insert_o((B),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o((B)),A2) ) ).

% subset_Compl_singleton
tff(fact_6107_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6108_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A2) ) ).

% subset_Compl_singleton
tff(fact_6109_ln__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_6110_ln__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_6111_ln__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% ln_gt_zero_iff
tff(fact_6112_ln__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( ( aa_real_real(ln_ln_real,Xa) = zero_zero_real )
      <=> ( Xa = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_6113_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_6114_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_6115_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_6116_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_simps(1)
tff(fact_6117_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_6118_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_6119_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).

% dbl_simps(1)
tff(fact_6120_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_simps(1)
tff(fact_6121_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_6122_add__neg__numeral__special_I7_J,axiom,
    aa_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_6123_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_6124_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_6125_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_6126_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_6127_add__neg__numeral__special_I8_J,axiom,
    aa_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_6128_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_6129_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_6130_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_6131_diff__numeral__special_I12_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_6132_diff__numeral__special_I12_J,axiom,
    aa_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_6133_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_6134_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_6135_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_6136_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6137_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6138_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6139_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6140_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6141_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6142_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6143_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6144_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6145_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6146_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).

% minus_one_mult_self
tff(fact_6147_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_6148_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).

% minus_one_mult_self
tff(fact_6149_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_6150_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_6151_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6152_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6153_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6154_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6155_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_6156_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_6157_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_6158_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(4)
tff(fact_6159_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(4)
tff(fact_6160_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(4)
tff(fact_6161_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_6162_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(3)
tff(fact_6163_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(3)
tff(fact_6164_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(3)
tff(fact_6165_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_6166_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(2)
tff(fact_6167_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).

% max_number_of(2)
tff(fact_6168_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(2)
tff(fact_6169_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_6170_ln__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_6171_ln__ge__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% ln_ge_zero_iff
tff(fact_6172_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6173_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6174_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6175_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_6176_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6177_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_6178_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_6179_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_6180_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_6181_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_6182_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_6183_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_6184_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Ma)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_6185_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Ma),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_6186_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_6187_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),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,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6188_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6189_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6190_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6191_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6192_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),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,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6193_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6194_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6195_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6196_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_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,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6197_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),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,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6198_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6199_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6200_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6201_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Y)) = aa_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,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6202_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_6203_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_6204_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Ma)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_6205_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_6206_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_6207_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_6208_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_6209_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_6210_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_6211_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_6212_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_6213_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_6214_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_6215_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_6216_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_6217_neg__numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Ma) ) ).

% neg_numeral_le_iff
tff(fact_6218_neg__numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Ma) ) ).

% neg_numeral_le_iff
tff(fact_6219_neg__numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Ma) ) ).

% neg_numeral_le_iff
tff(fact_6220_neg__numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Ma) ) ).

% neg_numeral_le_iff
tff(fact_6221_neg__numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Ma) ) ).

% neg_numeral_less_iff
tff(fact_6222_neg__numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Ma) ) ).

% neg_numeral_less_iff
tff(fact_6223_neg__numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Ma) ) ).

% neg_numeral_less_iff
tff(fact_6224_neg__numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Ma) ) ).

% neg_numeral_less_iff
tff(fact_6225_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Ma: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma)))
    <=> ( Ma != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6226_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Ma: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)))
    <=> ( Ma != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6227_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Ma: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma)))
    <=> ( Ma != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6228_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Ma: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)))
    <=> ( Ma != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6229_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_6230_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_6231_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_6232_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_6233_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6234_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6235_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6236_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6237_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6238_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6239_neg__numeral__less__neg__one__iff,axiom,
    ! [Ma: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Ma != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6240_neg__numeral__less__neg__one__iff,axiom,
    ! [Ma: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Ma != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6241_neg__numeral__less__neg__one__iff,axiom,
    ! [Ma: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Ma != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6242_neg__numeral__less__neg__one__iff,axiom,
    ! [Ma: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( Ma != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6243_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_6244_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_6245_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_6246_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_6247_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6248_power2__minus,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6249_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6250_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6251_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6252_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6253_add__neg__numeral__special_I9_J,axiom,
    aa_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_6254_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_6255_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6256_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6257_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6258_diff__numeral__special_I10_J,axiom,
    aa_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_6259_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_6260_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6261_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6262_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6263_diff__numeral__special_I11_J,axiom,
    aa_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_6264_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_6265_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6266_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6267_minus__1__div__2__eq,axiom,
    aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_6268_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_6269_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_6270_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_6271_minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_6272_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_6273_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6274_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6275_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6276_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6277_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6278_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6279_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6280_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6281_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6282_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6283_power__minus__odd,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_6284_power__minus__odd,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_6285_power__minus__odd,axiom,
    ! [Nb: nat,A: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_6286_power__minus__odd,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_6287_power__minus__odd,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_6288_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),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,Ma),one))) ).

% diff_numeral_special(4)
tff(fact_6289_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),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,Ma),one))) ).

% diff_numeral_special(4)
tff(fact_6290_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),one))) ).

% diff_numeral_special(4)
tff(fact_6291_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),one))) ).

% diff_numeral_special(4)
tff(fact_6292_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),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,Ma),one))) ).

% diff_numeral_special(4)
tff(fact_6293_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_6294_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_6295_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_6296_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_6297_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_6298_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_6299_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_6300_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_6301_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_6302_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_6303_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_6304_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_6305_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_6306_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_complex ).

% power_minus1_even
tff(fact_6307_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_6308_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_6309_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_6310_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_6311_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_6312_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_6313_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_6314_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_6315_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_6316_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_6317_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_6318_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_6319_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_6320_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_6321_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_6322_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_6323_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_6324_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_6325_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_6326_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_6327_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_6328_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_6329_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_6330_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_6331_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_6332_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_6333_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_6334_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_6335_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_6336_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_6337_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_6338_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_6339_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_6340_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_6341_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_6342_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_6343_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_6344_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_6345_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_6346_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_6347_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_6348_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_6349_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_6350_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_6351_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_6352_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_6353_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_6354_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_6355_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_6356_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_6357_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_6358_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_6359_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_num_real(numeral_numeral_real,Ma) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_6360_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_num_int(numeral_numeral_int,Ma) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_6361_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : numera6690914467698888265omplex(Ma) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_6362_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : numera6620942414471956472nteger(Ma) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_6363_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_num_rat(numeral_numeral_rat,Ma) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_6364_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma)) != aa_num_real(numeral_numeral_real,Nb) ).

% neg_numeral_neq_numeral
tff(fact_6365_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_6366_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_6367_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)) != numera6620942414471956472nteger(Nb) ).

% neg_numeral_neq_numeral
tff(fact_6368_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma)) != aa_num_rat(numeral_numeral_rat,Nb) ).

% neg_numeral_neq_numeral
tff(fact_6369_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_6370_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),A) = aa_int_int(times_times_int(B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_6371_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_6372_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_6373_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_6374_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_6375_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(times_times_int(A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_6376_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_6377_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_6378_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_6379_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_6380_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_6381_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_6382_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_6383_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_6384_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_6385_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_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_6386_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_6387_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_6388_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_6389_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_6390_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_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_6391_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_6392_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_6393_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_6394_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_6395_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_6396_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_6397_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_6398_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_6399_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% minus_diff_minus
tff(fact_6400_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A),B)) ).

% minus_diff_minus
tff(fact_6401_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) ).

% minus_diff_minus
tff(fact_6402_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_6403_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) ).

% minus_diff_minus
tff(fact_6404_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_6405_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_6406_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_6407_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_6408_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_6409_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_6410_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_6411_minus__divide__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_6412_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ).

% minus_divide_divide
tff(fact_6413_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).

% minus_divide_divide
tff(fact_6414_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ).

% minus_divide_divide
tff(fact_6415_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_6416_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_6417_minus__divide__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_divide_left
tff(fact_6418_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_6419_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% div_minus_right
tff(fact_6420_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_6421_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) ).

% mod_minus_right
tff(fact_6422_mod__minus__cong,axiom,
    ! [A: int,B: int,A3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = aa_int_int(modulo_modulo_int(A3),B) )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A3)),B) ) ) ).

% mod_minus_cong
tff(fact_6423_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A3,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3),B) ) ) ).

% mod_minus_cong
tff(fact_6424_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B))),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_6425_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% mod_minus_eq
tff(fact_6426_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_6427_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_6428_Collect__imp__eq,axiom,
    ! [P: fun_real_o,Q: fun_real_o] : collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ft(P),Q)) = sup_sup_set_real(uminus612125837232591019t_real(collect_real(P)),collect_real(Q)) ).

% Collect_imp_eq
tff(fact_6429_Collect__imp__eq,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_fu(P),Q)) = sup_sup_set_list_nat(uminus3195874150345416415st_nat(collect_list_nat(P)),collect_list_nat(Q)) ).

% Collect_imp_eq
tff(fact_6430_Collect__imp__eq,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_fv(P),Q)) = sup_sup_set_set_nat(uminus613421341184616069et_nat(collect_set_nat(P)),collect_set_nat(Q)) ).

% Collect_imp_eq
tff(fact_6431_Collect__imp__eq,axiom,
    ! [P: fun_int_o,Q: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_fw(P),Q)) = sup_sup_set_int(uminus1532241313380277803et_int(collect_int(P)),collect_int(Q)) ).

% Collect_imp_eq
tff(fact_6432_Collect__imp__eq,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_fx(P),Q)) = sup_sup_set_nat(uminus5710092332889474511et_nat(collect_nat(P)),collect_nat(Q)) ).

% Collect_imp_eq
tff(fact_6433_ln__less__self,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),Xa) ) ).

% ln_less_self
tff(fact_6434_neg__numeral__le__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_6435_neg__numeral__le__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_le_numeral
tff(fact_6436_neg__numeral__le__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_6437_neg__numeral__le__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_6438_not__numeral__le__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_6439_not__numeral__le__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_6440_not__numeral__le__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_6441_not__numeral__le__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_6442_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% zero_neq_neg_numeral
tff(fact_6443_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_6444_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_6445_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% zero_neq_neg_numeral
tff(fact_6446_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_neq_neg_numeral
tff(fact_6447_not__numeral__less__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_6448_not__numeral__less__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_6449_not__numeral__less__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_6450_not__numeral__less__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_6451_neg__numeral__less__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_less_numeral
tff(fact_6452_neg__numeral__less__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_6453_neg__numeral__less__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_less_numeral
tff(fact_6454_neg__numeral__less__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_less_numeral
tff(fact_6455_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_6456_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_6457_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_6458_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_6459_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_6460_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_6461_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_6462_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_6463_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6464_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(plus_plus_int(A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6465_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_6466_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6467_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6468_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6469_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(plus_plus_int(A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6470_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_6471_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6472_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6473_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_6474_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_6475_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_6476_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_6477_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_6478_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_6479_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_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_6480_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_6481_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_6482_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_6483_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_6484_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_6485_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_6486_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_6487_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_6488_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_6489_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_6490_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_6491_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_6492_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_6493_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_6494_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_6495_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_6496_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_6497_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_6498_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_6499_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_6500_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_6501_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_6502_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,Xa)) = aa_int_int(times_times_int(Xa),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_6503_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xa)) = aa_complex_complex(times_times_complex(Xa),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_6504_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xa)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_6505_numeral__times__minus__swap,axiom,
    ! [W: num,Xa: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,Xa)) = aa_rat_rat(times_times_rat(Xa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_6506_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6507_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_6508_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6509_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6510_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_6511_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6512_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_6513_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_6514_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_6515_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6620942414471956472nteger(Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_6516_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,Nb) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_6517_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% one_neq_neg_numeral
tff(fact_6518_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% one_neq_neg_numeral
tff(fact_6519_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_6520_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% one_neq_neg_numeral
tff(fact_6521_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_neq_neg_numeral
tff(fact_6522_square__eq__1__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa) = one_one_real )
    <=> ( ( Xa = one_one_real )
        | ( Xa = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_6523_square__eq__1__iff,axiom,
    ! [Xa: int] :
      ( ( aa_int_int(times_times_int(Xa),Xa) = one_one_int )
    <=> ( ( Xa = one_one_int )
        | ( Xa = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_6524_square__eq__1__iff,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_complex(times_times_complex(Xa),Xa) = one_one_complex )
    <=> ( ( Xa = one_one_complex )
        | ( Xa = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_6525_square__eq__1__iff,axiom,
    ! [Xa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xa) = one_one_Code_integer )
    <=> ( ( Xa = one_one_Code_integer )
        | ( Xa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_6526_square__eq__1__iff,axiom,
    ! [Xa: rat] :
      ( ( aa_rat_rat(times_times_rat(Xa),Xa) = one_one_rat )
    <=> ( ( Xa = one_one_rat )
        | ( Xa = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_6527_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6528_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6529_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_6530_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6531_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_6532_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_6533_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_6534_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_6535_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_6536_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_6537_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6538_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(plus_plus_int(K),B) )
     => ( aa_int_int(minus_minus_int(A),B2) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(minus_minus_int(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6539_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6540_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_6541_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_6542_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_6543_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_neg_div
tff(fact_6544_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_neg_div
tff(fact_6545_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_6546_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_neg_div
tff(fact_6547_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_6548_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_div_neg
tff(fact_6549_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_div_neg
tff(fact_6550_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_6551_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_div_neg
tff(fact_6552_inf__cancel__left1,axiom,
    ! [Xa: set_real,A: set_real,B: set_real] : inf_inf_set_real(inf_inf_set_real(Xa,A),inf_inf_set_real(uminus612125837232591019t_real(Xa),B)) = bot_bot_set_real ).

% inf_cancel_left1
tff(fact_6553_inf__cancel__left1,axiom,
    ! [Xa: set_o,A: set_o,B: set_o] : inf_inf_set_o(inf_inf_set_o(Xa,A),inf_inf_set_o(uminus_uminus_set_o(Xa),B)) = bot_bot_set_o ).

% inf_cancel_left1
tff(fact_6554_inf__cancel__left1,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] : inf_inf_set_nat(inf_inf_set_nat(Xa,A),inf_inf_set_nat(uminus5710092332889474511et_nat(Xa),B)) = bot_bot_set_nat ).

% inf_cancel_left1
tff(fact_6555_inf__cancel__left1,axiom,
    ! [Xa: set_int,A: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(Xa,A),inf_inf_set_int(uminus1532241313380277803et_int(Xa),B)) = bot_bot_set_int ).

% inf_cancel_left1
tff(fact_6556_inf__cancel__left2,axiom,
    ! [Xa: set_real,A: set_real,B: set_real] : inf_inf_set_real(inf_inf_set_real(uminus612125837232591019t_real(Xa),A),inf_inf_set_real(Xa,B)) = bot_bot_set_real ).

% inf_cancel_left2
tff(fact_6557_inf__cancel__left2,axiom,
    ! [Xa: set_o,A: set_o,B: set_o] : inf_inf_set_o(inf_inf_set_o(uminus_uminus_set_o(Xa),A),inf_inf_set_o(Xa,B)) = bot_bot_set_o ).

% inf_cancel_left2
tff(fact_6558_inf__cancel__left2,axiom,
    ! [Xa: set_nat,A: set_nat,B: set_nat] : inf_inf_set_nat(inf_inf_set_nat(uminus5710092332889474511et_nat(Xa),A),inf_inf_set_nat(Xa,B)) = bot_bot_set_nat ).

% inf_cancel_left2
tff(fact_6559_inf__cancel__left2,axiom,
    ! [Xa: set_int,A: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(uminus1532241313380277803et_int(Xa),A),inf_inf_set_int(Xa,B)) = bot_bot_set_int ).

% inf_cancel_left2
tff(fact_6560_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_6561_subset__Compl__self__eq,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),uminus_uminus_set_o(A2))
    <=> ( A2 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_6562_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_6563_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_6564_real__minus__mult__self__le,axiom,
    ! [U: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)) ).

% real_minus_mult_self_le
tff(fact_6565_Compl__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(sup_sup_set_nat(A2,B2)) = inf_inf_set_nat(uminus5710092332889474511et_nat(A2),uminus5710092332889474511et_nat(B2)) ).

% Compl_Un
tff(fact_6566_Compl__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : uminus5710092332889474511et_nat(inf_inf_set_nat(A2,B2)) = sup_sup_set_nat(uminus5710092332889474511et_nat(A2),uminus5710092332889474511et_nat(B2)) ).

% Compl_Int
tff(fact_6567_Diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = inf_inf_set_nat(A2,uminus5710092332889474511et_nat(B2)) ).

% Diff_eq
tff(fact_6568_zmult__eq__1__iff,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa_int_int(times_times_int(Ma),Nb) = one_one_int )
    <=> ( ( ( Ma = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Ma = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_6569_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa_int_int(times_times_int(Ma),Nb) = one_one_int )
     => ( ( Ma = one_one_int )
        | ( Ma = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_6570_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(minus_minus_int(zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_6571_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_6572_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_6573_minus__real__def,axiom,
    ! [Xa: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_6574_ln__one__minus__pos__upper__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xa))),aa_real_real(uminus_uminus_real,Xa)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_6575_ln__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),Xa) ) ).

% ln_bound
tff(fact_6576_ln__gt__zero__imp__gt__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_6577_ln__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_6578_ln__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa)) ) ).

% ln_gt_zero
tff(fact_6579_ln__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa)) ) ).

% ln_ge_zero
tff(fact_6580_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_6581_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_6582_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_6583_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_6584_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6585_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6586_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6587_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6588_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6589_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6590_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6591_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6592_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_6593_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_6594_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_6595_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_6596_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_6597_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_6598_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_6599_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_6600_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_6601_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_6602_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_6603_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_6604_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_6605_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_6606_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_6607_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_6608_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_6609_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_6610_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_6611_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_6612_not__one__le__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))) ).

% not_one_le_neg_numeral
tff(fact_6613_not__one__le__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))) ).

% not_one_le_neg_numeral
tff(fact_6614_not__one__le__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))) ).

% not_one_le_neg_numeral
tff(fact_6615_not__one__le__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))) ).

% not_one_le_neg_numeral
tff(fact_6616_not__numeral__le__neg__one,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_6617_not__numeral__le__neg__one,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_6618_not__numeral__le__neg__one,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_6619_not__numeral__le__neg__one,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_6620_neg__numeral__le__neg__one,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_6621_neg__numeral__le__neg__one,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_6622_neg__numeral__le__neg__one,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_6623_neg__numeral__le__neg__one,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_6624_neg__one__le__numeral,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Ma)) ).

% neg_one_le_numeral
tff(fact_6625_neg__one__le__numeral,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Ma)) ).

% neg_one_le_numeral
tff(fact_6626_neg__one__le__numeral,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Ma)) ).

% neg_one_le_numeral
tff(fact_6627_neg__one__le__numeral,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Ma)) ).

% neg_one_le_numeral
tff(fact_6628_neg__numeral__le__one,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),one_one_real) ).

% neg_numeral_le_one
tff(fact_6629_neg__numeral__le__one,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_6630_neg__numeral__le__one,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_6631_neg__numeral__le__one,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int) ).

% neg_numeral_le_one
tff(fact_6632_not__neg__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))) ).

% not_neg_one_less_neg_numeral
tff(fact_6633_not__neg__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))) ).

% not_neg_one_less_neg_numeral
tff(fact_6634_not__neg__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))) ).

% not_neg_one_less_neg_numeral
tff(fact_6635_not__neg__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))) ).

% not_neg_one_less_neg_numeral
tff(fact_6636_not__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))) ).

% not_one_less_neg_numeral
tff(fact_6637_not__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))) ).

% not_one_less_neg_numeral
tff(fact_6638_not__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))) ).

% not_one_less_neg_numeral
tff(fact_6639_not__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))) ).

% not_one_less_neg_numeral
tff(fact_6640_not__numeral__less__neg__one,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_6641_not__numeral__less__neg__one,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_6642_not__numeral__less__neg__one,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_6643_not__numeral__less__neg__one,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_6644_neg__one__less__numeral,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Ma)) ).

% neg_one_less_numeral
tff(fact_6645_neg__one__less__numeral,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Ma)) ).

% neg_one_less_numeral
tff(fact_6646_neg__one__less__numeral,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Ma)) ).

% neg_one_less_numeral
tff(fact_6647_neg__one__less__numeral,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Ma)) ).

% neg_one_less_numeral
tff(fact_6648_neg__numeral__less__one,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),one_one_real) ).

% neg_numeral_less_one
tff(fact_6649_neg__numeral__less__one,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int) ).

% neg_numeral_less_one
tff(fact_6650_neg__numeral__less__one,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_6651_neg__numeral__less__one,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_6652_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6653_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6654_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6655_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6656_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6657_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6658_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_6659_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_6660_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_6661_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_6662_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_6663_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_6664_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_6665_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_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_6666_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_6667_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_6668_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_6669_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_6670_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_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_6671_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_6672_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_6673_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_6674_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6675_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_6676_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6677_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_6678_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_6679_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_6680_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_6681_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_6682_power__minus,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_minus
tff(fact_6683_power__minus,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_minus
tff(fact_6684_power__minus,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_minus
tff(fact_6685_power__minus,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_minus
tff(fact_6686_power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_minus
tff(fact_6687_inf__shunt,axiom,
    ! [Xa: set_real,Y: set_real] :
      ( ( inf_inf_set_real(Xa,Y) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xa),uminus612125837232591019t_real(Y)) ) ).

% inf_shunt
tff(fact_6688_inf__shunt,axiom,
    ! [Xa: set_o,Y: set_o] :
      ( ( inf_inf_set_o(Xa,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(Xa),uminus_uminus_set_o(Y)) ) ).

% inf_shunt
tff(fact_6689_inf__shunt,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( inf_inf_set_int(Xa,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),uminus1532241313380277803et_int(Y)) ) ).

% inf_shunt
tff(fact_6690_inf__shunt,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat(Xa,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xa),uminus5710092332889474511et_nat(Y)) ) ).

% inf_shunt
tff(fact_6691_power__minus__Bit0,axiom,
    ! [Xa: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6692_power__minus__Bit0,axiom,
    ! [Xa: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6693_power__minus__Bit0,axiom,
    ! [Xa: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6694_power__minus__Bit0,axiom,
    ! [Xa: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6695_power__minus__Bit0,axiom,
    ! [Xa: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6696_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A2),uminus612125837232591019t_real(B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6697_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A2),uminus_uminus_set_o(B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6698_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int(A2,B2) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A2),uminus1532241313380277803et_int(B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6699_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat(A2,B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_6700_Compl__insert,axiom,
    ! [Xa: real,A2: set_real] : uminus612125837232591019t_real(insert_real(Xa,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real(Xa,bot_bot_set_real)) ).

% Compl_insert
tff(fact_6701_Compl__insert,axiom,
    ! [Xa: $o,A2: set_o] : uminus_uminus_set_o(insert_o((Xa),A2)) = minus_minus_set_o(uminus_uminus_set_o(A2),insert_o((Xa),bot_bot_set_o)) ).

% Compl_insert
tff(fact_6702_Compl__insert,axiom,
    ! [Xa: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(Xa,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int(Xa,bot_bot_set_int)) ).

% Compl_insert
tff(fact_6703_Compl__insert,axiom,
    ! [Xa: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(Xa,A2)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A2),insert_nat(Xa,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_6704_real__add__less__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xa)) ) ).

% real_add_less_0_iff
tff(fact_6705_real__0__less__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xa)),Y) ) ).

% real_0_less_add_iff
tff(fact_6706_real__0__le__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xa)),Y) ) ).

% real_0_le_add_iff
tff(fact_6707_real__add__le__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xa)) ) ).

% real_add_le_0_iff
tff(fact_6708_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(B),aa_int_int(modulo_modulo_int(A),B))) ).

% zmod_zminus1_eq_if
tff(fact_6709_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_6710_ln__ge__zero__imp__ge__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_6711_ln__add__one__self__le__self,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa) ) ).

% ln_add_one_self_le_self
tff(fact_6712_ln__mult,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_6713_ln__eq__minus__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( ( aa_real_real(ln_ln_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),one_one_real) )
       => ( Xa = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6714_ln__div,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_6715_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_6716_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_6717_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_6718_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_6719_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6720_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6721_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6722_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6723_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6724_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6725_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6726_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6727_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6728_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6729_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6730_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6731_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6732_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6733_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6734_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xa),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xa)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6735_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xa),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Xa)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6736_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6737_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6738_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6739_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6740_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xa: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xa),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xa)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6741_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xa),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xa)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6742_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6743_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6744_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6745_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6746_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6747_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6748_even__minus,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_6749_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_6750_power2__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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))) )
    <=> ( ( Xa = Y )
        | ( Xa = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6751_power2__eq__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xa),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))) )
    <=> ( ( Xa = Y )
        | ( Xa = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6752_power2__eq__iff,axiom,
    ! [Xa: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(Xa),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))) )
    <=> ( ( Xa = Y )
        | ( Xa = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6753_power2__eq__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),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))) )
    <=> ( ( Xa = Y )
        | ( Xa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6754_power2__eq__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),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))) )
    <=> ( ( Xa = Y )
        | ( Xa = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6755_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Nb))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(B2),Nb)),aa_int_int(divide_divide_int(A2),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_6756_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_6757_ln__le__minus__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_6758_ln__diff__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_6759_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6760_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6761_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6762_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6763_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6764_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6765_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6766_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6767_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_6768_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_6769_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_6770_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_6771_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6772_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6773_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6774_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6775_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_6776_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_6777_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_6778_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_6779_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_6780_uminus__power__if,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ).

% uminus_power_if
tff(fact_6781_uminus__power__if,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_int(power_power_int(A),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb))) ).

% uminus_power_if
tff(fact_6782_uminus__power__if,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_complex(power_power_complex(A),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb))) ).

% uminus_power_if
tff(fact_6783_uminus__power__if,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ).

% uminus_power_if
tff(fact_6784_uminus__power__if,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_rat(power_power_rat(A),Nb),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb))) ).

% uminus_power_if
tff(fact_6785_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6786_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6787_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6788_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6789_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6790_realpow__square__minus__le,axiom,
    ! [U: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_6791_ln__one__minus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xa)),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xa))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6792_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(L),one_one_int)),aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_6793_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(minus_minus_int(B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_6794_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_6795_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_6796_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R2))
     => ( ( B != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A),B,
            product_Pair_int_int($ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q2),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,Q2)),one_one_int)),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_6797_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6798_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6799_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6800_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6801_square__le__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_6802_square__le__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_6803_square__le__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_6804_square__le__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_6805_minus__power__mult__self,axiom,
    ! [A: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6806_minus__power__mult__self,axiom,
    ! [A: int,Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6807_minus__power__mult__self,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6808_minus__power__mult__self,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6809_minus__power__mult__self,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6810_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_6811_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_6812_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_6813_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_6814_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_6815_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_6816_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),L)),zero_zero_int)
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_6817_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_6818_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_6819_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_6820_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_6821_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_6822_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_6823_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_6824_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_6825_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K2: int] :
              ( aa_int_o(P,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(P,aa_int_int(times_times_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(P,K2)
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_6826_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_6827_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_ri631733984087533419it_int(Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_6828_option_Osize__gen_I1_J,axiom,
    ! [Xa: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(Xa,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_6829_option_Osize__gen_I1_J,axiom,
    ! [Xa: fun_nat_nat] : size_option_nat(Xa,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_6830_option_Osize__gen_I1_J,axiom,
    ! [Xa: fun_num_nat] : size_option_num(Xa,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_6831_ln__one__plus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6832_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_6833_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_6834_and__int_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xa),Xaa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_6835_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_6836_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6837_tanh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6838_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_6839_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),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,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_6840_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_6841_semiring__norm_I90_J,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_num(bit1,Ma) = aa_num_num(bit1,Nb) )
    <=> ( Ma = Nb ) ) ).

% semiring_norm(90)
tff(fact_6842_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_6843_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_6844_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_6845_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_6846_and_Oidem,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),A) = A ).

% and.idem
tff(fact_6847_and_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),A) = A ).

% and.idem
tff(fact_6848_and_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(A),B)) = aa_int_int(bit_se725231765392027082nd_int(A),B) ).

% and.left_idem
tff(fact_6849_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.left_idem
tff(fact_6850_and_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),B) = aa_int_int(bit_se725231765392027082nd_int(A),B) ).

% and.right_idem
tff(fact_6851_and_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.right_idem
tff(fact_6852_Compl__iff,axiom,
    ! [C: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(C),uminus8566677241136511917omplex(A2))
    <=> ~ aa_set_complex_o(member_complex(C),A2) ) ).

% Compl_iff
tff(fact_6853_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2))
    <=> ~ aa_set_real_o(member_real(C),A2) ) ).

% Compl_iff
tff(fact_6854_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2))
    <=> ~ aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% Compl_iff
tff(fact_6855_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2))
    <=> ~ aa_set_nat_o(member_nat(C),A2) ) ).

% Compl_iff
tff(fact_6856_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2))
    <=> ~ aa_set_int_o(member_int(C),A2) ) ).

% Compl_iff
tff(fact_6857_ComplI,axiom,
    ! [C: complex,A2: set_complex] :
      ( ~ aa_set_complex_o(member_complex(C),A2)
     => aa_set_complex_o(member_complex(C),uminus8566677241136511917omplex(A2)) ) ).

% ComplI
tff(fact_6858_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ aa_set_real_o(member_real(C),A2)
     => aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2)) ) ).

% ComplI
tff(fact_6859_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(C),A2)
     => aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2)) ) ).

% ComplI
tff(fact_6860_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(C),A2)
     => aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2)) ) ).

% ComplI
tff(fact_6861_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ aa_set_int_o(member_int(C),A2)
     => aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2)) ) ).

% ComplI
tff(fact_6862_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_6863_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_6864_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_6865_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_6866_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_6867_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_6868_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_6869_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_6870_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_6871_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_6872_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_6873_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_6874_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_6875_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_6876_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_6877_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_6878_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_6879_semiring__norm_I89_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(bit1,Ma) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_6880_semiring__norm_I88_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(bit0,Ma) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_6881_semiring__norm_I86_J,axiom,
    ! [Ma: num] : aa_num_num(bit1,Ma) != one ).

% semiring_norm(86)
tff(fact_6882_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_6883_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(Nb) ).

% abs_numeral
tff(fact_6884_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_numeral
tff(fact_6885_abs__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_numeral
tff(fact_6886_abs__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_numeral
tff(fact_6887_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_6888_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_6889_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_6890_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(times_times_int(A),A) ).

% abs_mult_self_eq
tff(fact_6891_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_6892_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_6893_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_6894_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_6895_abs__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_6896_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_6897_abs__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_divide
tff(fact_6898_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_6899_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_6900_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_6901_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_6902_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_6903_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_6904_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_6905_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_6906_bit_Oconj__zero__left,axiom,
    ! [Xa: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),Xa) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_6907_bit_Oconj__zero__right,axiom,
    ! [Xa: int] : aa_int_int(bit_se725231765392027082nd_int(Xa),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_6908_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_6909_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_6910_tanh__real__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(tanh_real,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_6911_semiring__norm_I73_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% semiring_norm(73)
tff(fact_6912_semiring__norm_I80_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% semiring_norm(80)
tff(fact_6913_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_6914_abs__of__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6915_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6916_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6917_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_6918_abs__le__self__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% abs_le_self_iff
tff(fact_6919_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_6920_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_6921_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_6922_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_6923_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_6924_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_6925_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_6926_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_6927_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_6928_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_6929_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_neg_numeral
tff(fact_6930_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_neg_numeral
tff(fact_6931_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(Nb) ).

% abs_neg_numeral
tff(fact_6932_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_neg_numeral
tff(fact_6933_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_6934_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_6935_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_6936_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_6937_abs__power__minus,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% abs_power_minus
tff(fact_6938_abs__power__minus,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) ).

% abs_power_minus
tff(fact_6939_abs__power__minus,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% abs_power_minus
tff(fact_6940_abs__power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% abs_power_minus
tff(fact_6941_and_Oleft__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),A) = A ).

% and.left_neutral
tff(fact_6942_and_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).

% and.left_neutral
tff(fact_6943_and_Oright__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_6944_and_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).

% and.right_neutral
tff(fact_6945_bit_Oconj__one__right,axiom,
    ! [Xa: code_integer] : bit_se3949692690581998587nteger(Xa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = Xa ).

% bit.conj_one_right
tff(fact_6946_bit_Oconj__one__right,axiom,
    ! [Xa: int] : aa_int_int(bit_se725231765392027082nd_int(Xa),aa_int_int(uminus_uminus_int,one_one_int)) = Xa ).

% bit.conj_one_right
tff(fact_6947_semiring__norm_I7_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% semiring_norm(7)
tff(fact_6948_semiring__norm_I9_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)) ).

% semiring_norm(9)
tff(fact_6949_semiring__norm_I14_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_6950_semiring__norm_I15_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Ma)),Nb)) ).

% semiring_norm(15)
tff(fact_6951_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_6952_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_6953_semiring__norm_I72_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% semiring_norm(72)
tff(fact_6954_semiring__norm_I81_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% semiring_norm(81)
tff(fact_6955_semiring__norm_I70_J,axiom,
    ! [Ma: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Ma)),one) ).

% semiring_norm(70)
tff(fact_6956_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_6957_tanh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_6958_tanh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% tanh_real_pos_iff
tff(fact_6959_tanh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_6960_tanh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% tanh_real_nonneg_iff
tff(fact_6961_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6962_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6963_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6964_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6965_abs__of__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_6966_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_6967_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_6968_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_6969_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = one_one_int ).

% and_numerals(2)
tff(fact_6970_and__numerals_I2_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(bit1,Y))) = one_one_nat ).

% and_numerals(2)
tff(fact_6971_and__numerals_I8_J,axiom,
    ! [Xa: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_6972_and__numerals_I8_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_6973_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_6974_semiring__norm_I10_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)),one)) ).

% semiring_norm(10)
tff(fact_6975_semiring__norm_I8_J,axiom,
    ! [Ma: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Ma)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),one)) ).

% semiring_norm(8)
tff(fact_6976_semiring__norm_I5_J,axiom,
    ! [Ma: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Ma)),one) = aa_num_num(bit1,Ma) ).

% semiring_norm(5)
tff(fact_6977_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% semiring_norm(4)
tff(fact_6978_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_6979_semiring__norm_I16_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,Ma),Nb)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Ma),Nb)))) ).

% semiring_norm(16)
tff(fact_6980_semiring__norm_I74_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% semiring_norm(74)
tff(fact_6981_semiring__norm_I79_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% semiring_norm(79)
tff(fact_6982_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6983_zero__less__power__abs__iff,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb))
    <=> ( ( A != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6984_zero__less__power__abs__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb))
    <=> ( ( A != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6985_zero__less__power__abs__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb))
    <=> ( ( A != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6986_and__numerals_I5_J,axiom,
    ! [Xa: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_6987_and__numerals_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_6988_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_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_6989_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_6990_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6991_abs__power2,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6992_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6993_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6994_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6995_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6996_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6997_power2__abs,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6998_and__numerals_I3_J,axiom,
    ! [Xa: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(3)
tff(fact_6999_and__numerals_I3_J,axiom,
    ! [Xa: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),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,Xa)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(3)
tff(fact_7000_and__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_7001_and__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_7002_and__numerals_I6_J,axiom,
    ! [Xa: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(6)
tff(fact_7003_and__numerals_I6_J,axiom,
    ! [Xa: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),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,Xa)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(6)
tff(fact_7004_and__numerals_I4_J,axiom,
    ! [Xa: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(4)
tff(fact_7005_and__numerals_I4_J,axiom,
    ! [Xa: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),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,Xa)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(4)
tff(fact_7006_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7007_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7008_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7009_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7010_div__Suc__eq__div__add3,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_7011_Suc__div__eq__add3__div__numeral,axiom,
    ! [Ma: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Ma)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Ma)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_7012_mod__Suc__eq__mod__add3,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_7013_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Ma: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Ma)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Ma)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_7014_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_7015_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_7016_and__numerals_I7_J,axiom,
    ! [Xa: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_7017_and__numerals_I7_J,axiom,
    ! [Xa: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),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,Xa)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_7018_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_7019_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_7020_ComplD,axiom,
    ! [C: complex,A2: set_complex] :
      ( aa_set_complex_o(member_complex(C),uminus8566677241136511917omplex(A2))
     => ~ aa_set_complex_o(member_complex(C),A2) ) ).

% ComplD
tff(fact_7021_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( aa_set_real_o(member_real(C),uminus612125837232591019t_real(A2))
     => ~ aa_set_real_o(member_real(C),A2) ) ).

% ComplD
tff(fact_7022_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C),uminus613421341184616069et_nat(A2))
     => ~ aa_set_set_nat_o(member_set_nat(C),A2) ) ).

% ComplD
tff(fact_7023_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( aa_set_nat_o(member_nat(C),uminus5710092332889474511et_nat(A2))
     => ~ aa_set_nat_o(member_nat(C),A2) ) ).

% ComplD
tff(fact_7024_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( aa_set_int_o(member_int(C),uminus1532241313380277803et_int(A2))
     => ~ aa_set_int_o(member_int(C),A2) ) ).

% ComplD
tff(fact_7025_Compl__eq,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(aTP_Lamm_fy(A2)) ).

% Compl_eq
tff(fact_7026_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_fz(A2)) ).

% Compl_eq
tff(fact_7027_Compl__eq,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(aTP_Lamm_ga(A2)) ).

% Compl_eq
tff(fact_7028_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_gb(A2)) ).

% Compl_eq
tff(fact_7029_Compl__eq,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(aTP_Lamm_gc(A2)) ).

% Compl_eq
tff(fact_7030_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_gd(A2)) ).

% Compl_eq
tff(fact_7031_Collect__neg__eq,axiom,
    ! [P: fun_real_o] : collect_real(aTP_Lamm_ge(P)) = uminus612125837232591019t_real(collect_real(P)) ).

% Collect_neg_eq
tff(fact_7032_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_o] : collect_list_nat(aTP_Lamm_gf(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_7033_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_o] : collect_set_nat(aTP_Lamm_gg(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_7034_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : collect_nat(aTP_Lamm_gh(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_7035_Collect__neg__eq,axiom,
    ! [P: fun_int_o] : collect_int(aTP_Lamm_gi(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_7036_uminus__set__def,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(uminus4035047602201070266plex_o(aTP_Lamm_a(A2))) ).

% uminus_set_def
tff(fact_7037_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_aa(A2))) ).

% uminus_set_def
tff(fact_7038_uminus__set__def,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(uminus8676031017987022600_nat_o(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_7039_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_7040_uminus__set__def,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ad(A2))) ).

% uminus_set_def
tff(fact_7041_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_7042_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),C) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B),C)) ).

% and.assoc
tff(fact_7043_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.assoc
tff(fact_7044_and_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(bit_se725231765392027082nd_int(B),A) ).

% and.commute
tff(fact_7045_and_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),A) ).

% and.commute
tff(fact_7046_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(B),aa_int_int(bit_se725231765392027082nd_int(A),C)) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B),C)) ).

% and.left_commute
tff(fact_7047_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.left_commute
tff(fact_7048_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_7049_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_7050_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_7051_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_7052_abs__ge__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).

% abs_ge_self
tff(fact_7053_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_7054_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_self
tff(fact_7055_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_7056_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_7057_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_7058_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_7059_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_7060_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_7061_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_7062_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_7063_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_7064_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_7065_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_7066_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% abs_minus_commute
tff(fact_7067_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_7068_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B)) = aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),A)) ).

% abs_minus_commute
tff(fact_7069_power__abs,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) ).

% power_abs
tff(fact_7070_power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) ).

% power_abs
tff(fact_7071_power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) ).

% power_abs
tff(fact_7072_power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) ).

% power_abs
tff(fact_7073_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa_num_num(bit0,X2) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_7074_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_7075_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_7076_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_7077_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_7078_abs__ge__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).

% abs_ge_zero
tff(fact_7079_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_zero
tff(fact_7080_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_7081_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_7082_abs__not__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_7083_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_7084_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_7085_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_7086_abs__of__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_7087_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_7088_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_7089_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_7090_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_7091_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_7092_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(A),B))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_7093_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_7094_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)) ) ) ).

% abs_mult_less
tff(fact_7095_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_7096_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(times_times_int(C),D)) ) ) ).

% abs_mult_less
tff(fact_7097_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_7098_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_7099_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_7100_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_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(minus_minus_int(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_7101_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_7102_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq3
tff(fact_7103_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq3
tff(fact_7104_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(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(minus_minus_int(A),B))) ).

% abs_triangle_ineq3
tff(fact_7105_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_7106_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq2
tff(fact_7107_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq2
tff(fact_7108_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_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(minus_minus_int(A),B))) ).

% abs_triangle_ineq2
tff(fact_7109_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_7110_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_7111_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_7112_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_7113_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).

% abs_leI
tff(fact_7114_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_7115_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_7116_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_7117_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_7118_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_7119_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_7120_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_7121_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_7122_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_7123_abs__ge__minus__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).

% abs_ge_minus_self
tff(fact_7124_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_7125_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_minus_self
tff(fact_7126_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_7127_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_less_iff
tff(fact_7128_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_7129_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_less_iff
tff(fact_7130_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_7131_AND__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(Xa),Y)) ) ).

% AND_lower
tff(fact_7132_AND__upper1,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xa),Y)),Xa) ) ).

% AND_upper1
tff(fact_7133_AND__upper2,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xa),Y)),Y) ) ).

% AND_upper2
tff(fact_7134_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_7135_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xa),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_7136_abs__real__def,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_real_def
tff(fact_7137_xor__num_Ocases,axiom,
    ! [Xa: product_prod_num_num] :
      ( ( Xa != product_Pair_num_num(one,one) )
     => ( ! [N: num] : Xa != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : Xa != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M2: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M2),one)
           => ( ! [M2: num,N: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit0,N))
             => ( ! [M2: num,N: num] : Xa != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit1,N))
               => ( ! [M2: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M2),one)
                 => ( ! [M2: num,N: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit0,N))
                   => ~ ! [M2: num,N: num] : Xa != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_7138_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_7139_dense__eq0__I,axiom,
    ! [Xa: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),E) )
     => ( Xa = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_7140_dense__eq0__I,axiom,
    ! [Xa: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),E) )
     => ( Xa = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_7141_abs__mult__pos,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),Xa) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Xa)) ) ) ).

% abs_mult_pos
tff(fact_7142_abs__mult__pos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),Xa) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xa)) ) ) ).

% abs_mult_pos
tff(fact_7143_abs__mult__pos,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y)),Xa) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),Xa)) ) ) ).

% abs_mult_pos
tff(fact_7144_abs__mult__pos,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_int(times_times_int(aa_int_int(abs_abs_int,Y)),Xa) = aa_int_int(abs_abs_int,aa_int_int(times_times_int(Y),Xa)) ) ) ).

% abs_mult_pos
tff(fact_7145_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_7146_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_7147_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_7148_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_7149_abs__minus__le__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_7150_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_7151_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_7152_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_7153_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7154_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7155_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7156_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7157_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7158_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7159_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7160_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7161_abs__div__pos,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,Xa)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) ) ) ).

% abs_div_pos
tff(fact_7162_abs__div__pos,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xa)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xa),Y)) ) ) ).

% abs_div_pos
tff(fact_7163_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).

% zero_le_power_abs
tff(fact_7164_zero__le__power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) ).

% zero_le_power_abs
tff(fact_7165_zero__le__power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb)) ).

% zero_le_power_abs
tff(fact_7166_zero__le__power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb)) ).

% zero_le_power_abs
tff(fact_7167_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_7168_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_7169_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_7170_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_7171_abs__of__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_7172_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_7173_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_7174_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_7175_abs__if__raw,axiom,
    ! [X: real] :
      aa_real_real(abs_abs_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real),aa_real_real(uminus_uminus_real,X),X) ).

% abs_if_raw
tff(fact_7176_abs__if__raw,axiom,
    ! [X: int] :
      aa_int_int(abs_abs_int,X) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int),aa_int_int(uminus_uminus_int,X),X) ).

% abs_if_raw
tff(fact_7177_abs__if__raw,axiom,
    ! [X: code_integer] :
      abs_abs_Code_integer(X) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),X) ).

% abs_if_raw
tff(fact_7178_abs__if__raw,axiom,
    ! [X: rat] :
      aa_rat_rat(abs_abs_rat,X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X),X) ).

% abs_if_raw
tff(fact_7179_abs__diff__le__iff,axiom,
    ! [Xa: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xa,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7180_abs__diff__le__iff,axiom,
    ! [Xa: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7181_abs__diff__le__iff,axiom,
    ! [Xa: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xa),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7182_abs__diff__le__iff,axiom,
    ! [Xa: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xa),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),R2)),Xa)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_int_int(plus_plus_int(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_7183_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_7184_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_7185_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_7186_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_7187_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_7188_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7189_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7190_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),aa_int_int(plus_plus_int(C),D)))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),C))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7191_abs__diff__less__iff,axiom,
    ! [Xa: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xa,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7192_abs__diff__less__iff,axiom,
    ! [Xa: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7193_abs__diff__less__iff,axiom,
    ! [Xa: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xa),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7194_abs__diff__less__iff,axiom,
    ! [Xa: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xa),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),R2)),Xa)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_int_int(plus_plus_int(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_7195_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),K) ) ).

% and_less_eq
tff(fact_7196_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_7197_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Xa),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_7198_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit1,Nb)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Nb),aa_num_Extended_enat(numera1916890842035813515d_enat,Nb)),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_7199_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_7200_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_7201_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_7202_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_7203_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_7204_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_7205_cong__exp__iff__simps_I13_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_7206_cong__exp__iff__simps_I13_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_7207_cong__exp__iff__simps_I13_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_7208_cong__exp__iff__simps_I12_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_7209_cong__exp__iff__simps_I12_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_7210_cong__exp__iff__simps_I12_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_7211_cong__exp__iff__simps_I10_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_7212_cong__exp__iff__simps_I10_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_7213_cong__exp__iff__simps_I10_J,axiom,
    ! [Ma: num,Q2: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_7214_power__minus__Bit1,axiom,
    ! [Xa: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7215_power__minus__Bit1,axiom,
    ! [Xa: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7216_power__minus__Bit1,axiom,
    ! [Xa: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7217_power__minus__Bit1,axiom,
    ! [Xa: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7218_power__minus__Bit1,axiom,
    ! [Xa: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7219_lemma__interval__lt,axiom,
    ! [A: real,Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y4))),D3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_7220_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(bit1,Nb)) = $let(
        m2: extended_enat,
        m2:= aa_num_Extended_enat(numera1916890842035813515d_enat,Nb),
        plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(m2,m2),one_on7984719198319812577d_enat) ) ).

% numeral_code(3)
tff(fact_7221_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_7222_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_7223_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_7224_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_7225_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_7226_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7227_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7228_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7229_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7230_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(times_times_int(aa_int_int(times_times_int(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7231_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3949692690581998587nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_7232_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_7233_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_7234_abs__add__one__gt__zero,axiom,
    ! [Xa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xa))) ).

% abs_add_one_gt_zero
tff(fact_7235_abs__add__one__gt__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xa))) ).

% abs_add_one_gt_zero
tff(fact_7236_abs__add__one__gt__zero,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,Xa))) ).

% abs_add_one_gt_zero
tff(fact_7237_abs__add__one__gt__zero,axiom,
    ! [Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),aa_int_int(abs_abs_int,Xa))) ).

% abs_add_one_gt_zero
tff(fact_7238_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).

% even_and_iff_int
tff(fact_7239_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit1_div_2
tff(fact_7240_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_7241_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_7242_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_7243_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_7244_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_7245_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_7246_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_7247_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_7248_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_7249_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),A)),A) ).

% power3_eq_cube
tff(fact_7250_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_7251_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(times_times_int(aa_int_int(times_times_int(A),A)),A) ).

% power3_eq_cube
tff(fact_7252_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_7253_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_7254_lemma__interval,axiom,
    ! [A: real,Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y4))),D3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_7255_mod__exhaust__less__4,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),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,Ma),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,Ma),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,Ma),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_7256_and__one__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,one_one_Code_integer) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_7257_and__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),one_one_int) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_7258_and__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_7259_one__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_7260_one__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_7261_one__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_7262_abs__le__square__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xa)),abs_abs_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),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_7263_abs__le__square__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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_7264_abs__le__square__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),aa_rat_rat(abs_abs_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),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_7265_abs__le__square__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),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_7266_abs__square__eq__1,axiom,
    ! [Xa: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xa) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_7267_abs__square__eq__1,axiom,
    ! [Xa: rat] :
      ( ( aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,Xa) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_7268_abs__square__eq__1,axiom,
    ! [Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,Xa) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_7269_abs__square__eq__1,axiom,
    ! [Xa: int] :
      ( ( aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xa) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_7270_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_7271_num_Osize_I6_J,axiom,
    ! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_7272_power__even__abs,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% power_even_abs
tff(fact_7273_power__even__abs,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_even_abs
tff(fact_7274_power__even__abs,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_even_abs
tff(fact_7275_power__even__abs,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_even_abs
tff(fact_7276_cong__exp__iff__simps_I11_J,axiom,
    ! [Ma: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),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,Ma)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7277_cong__exp__iff__simps_I11_J,axiom,
    ! [Ma: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_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(modulo_modulo_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7278_cong__exp__iff__simps_I11_J,axiom,
    ! [Ma: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7279_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7280_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7281_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7282_Suc__div__eq__add3__div,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Ma)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Ma)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_7283_Suc__mod__eq__add3__mod,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Ma)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Ma)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_7284_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xa: code_integer] :
      ( ! [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X3),aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7285_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xa: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
         => aa_real_o(aa_real_fun_real_o(P,X3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7286_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xa: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
         => aa_rat_o(aa_rat_fun_rat_o(P,X3),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,Xa)),aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7287_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xa: int] :
      ( ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_int_o(aa_int_fun_int_o(P,X3),aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xa)),aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7288_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xa)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7289_power2__le__iff__abs__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7290_power2__le__iff__abs__le,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7291_power2__le__iff__abs__le,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7292_abs__square__le__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xa)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_7293_abs__square__le__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real) ) ).

% abs_square_le_1
tff(fact_7294_abs__square__le__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xa)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_7295_abs__square__le__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xa)),one_one_int) ) ).

% abs_square_le_1
tff(fact_7296_abs__square__less__1,axiom,
    ! [Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xa)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_7297_abs__square__less__1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real) ) ).

% abs_square_less_1
tff(fact_7298_abs__square__less__1,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xa)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_7299_abs__square__less__1,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xa)),one_one_int) ) ).

% abs_square_less_1
tff(fact_7300_power__mono__even,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7301_power__mono__even,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_even
tff(fact_7302_power__mono__even,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7303_power__mono__even,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7304_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_7305_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( ( K = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_7306_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_7307_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_7308_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_7309_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_7310_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_7311_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_7312_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7313_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_7314_divmod__algorithm__code_I7_J,axiom,
    ! [Ma: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_7315_divmod__algorithm__code_I7_J,axiom,
    ! [Ma: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),unique5024387138958732305ep_int(aa_num_num(bit1,Nb),unique5052692396658037445od_int(aa_num_num(bit0,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_7316_divmod__algorithm__code_I7_J,axiom,
    ! [Ma: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,Ma))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_7317_divmod__algorithm__code_I8_J,axiom,
    ! [Ma: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_7318_divmod__algorithm__code_I8_J,axiom,
    ! [Ma: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),unique5024387138958732305ep_int(aa_num_num(bit1,Nb),unique5052692396658037445od_int(aa_num_num(bit1,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_7319_divmod__algorithm__code_I8_J,axiom,
    ! [Ma: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,Ma))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_7320_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_7321_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K,L))
     => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
            ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_7322_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_7323_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_7324_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_7325_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_7326_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_7327_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_7328_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Nb = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_7329_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_7330_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_7331_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_7332_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% less_numeral_Suc
tff(fact_7333_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_7334_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_7335_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_7336_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% le_numeral_Suc
tff(fact_7337_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_7338_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Nb) ).

% diff_numeral_Suc
tff(fact_7339_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_7340_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% max_numeral_Suc
tff(fact_7341_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_7342_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_7343_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_7344_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_7345_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_7346_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_7347_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_7348_dvd__numeral__simp,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))
    <=> unique6319869463603278526ux_int(unique5052692396658037445od_int(Nb,Ma)) ) ).

% dvd_numeral_simp
tff(fact_7349_dvd__numeral__simp,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> unique6322359934112328802ux_nat(unique5055182867167087721od_nat(Nb,Ma)) ) ).

% dvd_numeral_simp
tff(fact_7350_dvd__numeral__simp,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(Ma)),numera6620942414471956472nteger(Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Ma)) ) ).

% dvd_numeral_simp
tff(fact_7351_divmod__algorithm__code_I2_J,axiom,
    ! [Ma: num] : unique5052692396658037445od_int(Ma,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,Ma),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_7352_divmod__algorithm__code_I2_J,axiom,
    ! [Ma: num] : unique5055182867167087721od_nat(Ma,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,Ma)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_7353_divmod__algorithm__code_I2_J,axiom,
    ! [Ma: num] : unique3479559517661332726nteger(Ma,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(Ma),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_7354_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_7355_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_7356_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_7357_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_7358_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_7359_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_7360_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_7361_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_7362_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_7363_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_7364_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_7365_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_7366_abs__zmult__eq__1,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(times_times_int(Ma),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Ma) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7367_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_7368_zabs__def,axiom,
    ! [Ia: int] :
      aa_int_int(abs_abs_int,Ia) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),zero_zero_int),aa_int_int(uminus_uminus_int,Ia),Ia) ).

% zabs_def
tff(fact_7369_dvd__imp__le__int,axiom,
    ! [Ia: int,D: int] :
      ( ( Ia != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Ia)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,Ia)) ) ) ).

% dvd_imp_le_int
tff(fact_7370_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_7371_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_7372_zdvd__mult__cancel1,axiom,
    ! [Ma: int,Nb: int] :
      ( ( Ma != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(Ma),Nb)),Ma)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7373_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,K)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_abs_add_iff
tff(fact_7374_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_add_abs_iff
tff(fact_7375_divmod__int__def,axiom,
    ! [Ma: num,Nb: num] : unique5052692396658037445od_int(Ma,Nb) = product_Pair_int_int(aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_7376_divmod__def,axiom,
    ! [Ma: num,Nb: num] : unique5052692396658037445od_int(Ma,Nb) = product_Pair_int_int(aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_7377_divmod__def,axiom,
    ! [Ma: num,Nb: num] : unique5055182867167087721od_nat(Ma,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_7378_divmod__def,axiom,
    ! [Ma: num,Nb: num] : unique3479559517661332726nteger(Ma,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb))) ).

% divmod_def
tff(fact_7379_divmod_H__nat__def,axiom,
    ! [Ma: num,Nb: num] : unique5055182867167087721od_nat(Ma,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_7380_nat__intermed__int__val,axiom,
    ! [Ma: nat,Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Ma)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7381_dbl__dec__def,axiom,
    ! [Xa: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xa) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(Xa),Xa)),one_one_complex) ).

% dbl_dec_def
tff(fact_7382_dbl__dec__def,axiom,
    ! [Xa: real] : aa_real_real(neg_nu6075765906172075777c_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Xa)),one_one_real) ).

% dbl_dec_def
tff(fact_7383_dbl__dec__def,axiom,
    ! [Xa: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xa) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Xa),Xa)),one_one_rat) ).

% dbl_dec_def
tff(fact_7384_dbl__dec__def,axiom,
    ! [Xa: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xa) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(Xa),Xa)),one_one_int) ).

% dbl_dec_def
tff(fact_7385_decr__lemma,axiom,
    ! [D: int,Xa: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(Xa),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xa),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_7386_incr__lemma,axiom,
    ! [D: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(plus_plus_int(Xa),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xa),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_7387_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7388_and__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Ma),Nb) = $ite(
        ( ( Ma = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_7389_and__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_7390_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7391_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(A0,A1))
     => ( ! [K2: int,L2: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K2,L2))
           => ( ( ~ ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_7392_divmod__divmod__step,axiom,
    ! [Ma: num,Nb: num] :
      unique5055182867167087721od_nat(Ma,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Ma)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Ma,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_7393_divmod__divmod__step,axiom,
    ! [Ma: num,Nb: num] :
      unique5052692396658037445od_int(Ma,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,Ma)),unique5024387138958732305ep_int(Nb,unique5052692396658037445od_int(Ma,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_7394_divmod__divmod__step,axiom,
    ! [Ma: num,Nb: num] :
      unique3479559517661332726nteger(Ma,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Ma)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Ma,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_7395_and__int_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xa),Xaa) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(Xa,Xaa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(Xa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(Xa,Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_7396_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I2: int,J3: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I2,J3))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J3)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(plus_plus_int(I2),one_one_int)),J3) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_7397_arctan__double,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),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,Xa)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_7398_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_7399_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_7400_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_7401_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7402_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(
        K = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: real,
            l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_7403_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(
        K = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: int,
            l:= aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(plus_plus_int(l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_7404_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_complex(ring_17405671764205052669omplex,K) = $ite(
        K = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_7405_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K) = $ite(
        K = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_7406_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(
        K = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(plus_plus_rat(l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_7407_divmod__algorithm__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Ma),aa_num_num(bit0,Nb)) = produc4245557441103728435nt_int(aTP_Lamm_gj,unique5052692396658037445od_int(Ma,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_7408_divmod__algorithm__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Ma),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gk),unique5055182867167087721od_nat(Ma,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_7409_divmod__algorithm__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit1,Ma),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_gl,unique3479559517661332726nteger(Ma,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_7410_arctan__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(arctan,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_7411_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_7412_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_7413_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_7414_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_7415_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_7416_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_7417_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_7418_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_7419_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_7420_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_7421_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_7422_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_7423_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_7424_of__int__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_7425_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_7426_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_7427_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_7428_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_7429_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_7430_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_7431_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_7432_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_7433_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_7434_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_7435_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(times_times_int(W),Z)) = aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_mult
tff(fact_7436_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(times_times_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_7437_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(times_times_int(W),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_7438_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(times_times_int(W),Z)) = aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_7439_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(W),Z)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_7440_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(plus_plus_int(W),Z)) = aa_int_int(plus_plus_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_7441_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_7442_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(plus_plus_int(W),Z)) = aa_complex_complex(plus_plus_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_add
tff(fact_7443_arctan__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_7444_zero__less__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% zero_less_arctan_iff
tff(fact_7445_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_7446_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_7447_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).

% of_int_power
tff(fact_7448_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,Xa) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_7449_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,Xa) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_7450_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,Xa) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xa ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_7451_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xa) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( Xa = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_7452_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xa) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( Xa = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_7453_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xa) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( Xa = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_7454_zero__le__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% zero_le_arctan_iff
tff(fact_7455_arctan__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_7456_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_7457_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_7458_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_7459_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_7460_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_7461_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_7462_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_7463_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_7464_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_7465_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7466_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7467_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7468_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7469_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_dec_simps(1)
tff(fact_7470_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_dec_simps(1)
tff(fact_7471_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).

% dbl_dec_simps(1)
tff(fact_7472_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K))) ).

% dbl_dec_simps(1)
tff(fact_7473_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_dec_simps(1)
tff(fact_7474_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_inc_simps(1)
tff(fact_7475_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_inc_simps(1)
tff(fact_7476_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).

% dbl_inc_simps(1)
tff(fact_7477_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K))) ).

% dbl_inc_simps(1)
tff(fact_7478_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_inc_simps(1)
tff(fact_7479_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_7480_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_7481_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_7482_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_7483_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_7484_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_7485_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_7486_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_7487_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_7488_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_7489_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_7490_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_7491_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_7492_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_7493_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_7494_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_7495_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_7496_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_7497_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_7498_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_7499_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_7500_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_7501_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_7502_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_7503_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_7504_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_7505_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_7506_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_7507_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_7508_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_7509_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xa)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_7510_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_7511_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_7512_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_7513_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xa)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_7514_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_7515_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_7516_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_7517_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_7518_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_7519_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_7520_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_7521_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xa)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_7522_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xa)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_7523_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_7524_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_7525_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,Xa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xa) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_7526_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_7527_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xa)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_7528_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xa: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xa)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_7529_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7530_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7531_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_7532_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7533_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7534_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_7535_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7536_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7537_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_7538_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7539_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7540_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_7541_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7542_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7543_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xa))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7544_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7545_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7546_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7547_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7548_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xa))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7549_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa))),Nb) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7550_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7551_divmod__algorithm__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Ma),aa_num_num(bit0,Nb)) = produc4245557441103728435nt_int(aTP_Lamm_gm,unique5052692396658037445od_int(Ma,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_7552_divmod__algorithm__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Ma),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gn),unique5055182867167087721od_nat(Ma,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_7553_divmod__algorithm__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit0,Ma),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_go,unique3479559517661332726nteger(Ma,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_7554_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7555_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7556_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7557_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7558_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7559_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7560_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7561_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7562_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7563_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7564_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7565_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7566_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xa))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7567_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7568_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xa))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7569_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xa))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xa))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7570_mult__of__int__commute,axiom,
    ! [Xa: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Xa)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,Xa)) ).

% mult_of_int_commute
tff(fact_7571_mult__of__int__commute,axiom,
    ! [Xa: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Xa)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,Xa)) ).

% mult_of_int_commute
tff(fact_7572_mult__of__int__commute,axiom,
    ! [Xa: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,Xa)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,Xa)) ).

% mult_of_int_commute
tff(fact_7573_mult__of__int__commute,axiom,
    ! [Xa: int,Y: int] : aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,Xa)),Y) = aa_int_int(times_times_int(Y),aa_int_int(ring_1_of_int_int,Xa)) ).

% mult_of_int_commute
tff(fact_7574_of__int__and__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se725231765392027082nd_int(K),L)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_and_eq
tff(fact_7575_real__of__int__div4,axiom,
    ! [Nb: int,Xa: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xa))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xa))) ).

% real_of_int_div4
tff(fact_7576_real__of__int__div,axiom,
    ! [D: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Nb)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_7577_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_nonneg
tff(fact_7578_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_nonneg
tff(fact_7579_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_7580_of__int__leD,axiom,
    ! [Nb: int,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xa)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xa) ) ) ).

% of_int_leD
tff(fact_7581_of__int__leD,axiom,
    ! [Nb: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xa)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% of_int_leD
tff(fact_7582_of__int__leD,axiom,
    ! [Nb: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xa)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xa) ) ) ).

% of_int_leD
tff(fact_7583_of__int__leD,axiom,
    ! [Nb: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xa)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xa) ) ) ).

% of_int_leD
tff(fact_7584_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_pos
tff(fact_7585_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_7586_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_7587_of__int__lessD,axiom,
    ! [Nb: int,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xa)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xa) ) ) ).

% of_int_lessD
tff(fact_7588_of__int__lessD,axiom,
    ! [Nb: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xa)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% of_int_lessD
tff(fact_7589_of__int__lessD,axiom,
    ! [Nb: int,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xa)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xa) ) ) ).

% of_int_lessD
tff(fact_7590_of__int__lessD,axiom,
    ! [Nb: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xa)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xa) ) ) ).

% of_int_lessD
tff(fact_7591_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K)) ).

% of_int_neg_numeral
tff(fact_7592_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_7593_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_7594_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K)) ).

% of_int_neg_numeral
tff(fact_7595_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).

% of_int_neg_numeral
tff(fact_7596_int__le__real__less,axiom,
    ! [Nb: int,Ma: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Ma)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Ma)),one_one_real)) ) ).

% int_le_real_less
tff(fact_7597_int__less__real__le,axiom,
    ! [Nb: int,Ma: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Ma)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Ma)) ) ).

% int_less_real_le
tff(fact_7598_dbl__inc__def,axiom,
    ! [Xa: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,Xa) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(Xa),Xa)),one_one_complex) ).

% dbl_inc_def
tff(fact_7599_dbl__inc__def,axiom,
    ! [Xa: real] : aa_real_real(neg_nu8295874005876285629c_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Xa)),one_one_real) ).

% dbl_inc_def
tff(fact_7600_dbl__inc__def,axiom,
    ! [Xa: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,Xa) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Xa),Xa)),one_one_rat) ).

% dbl_inc_def
tff(fact_7601_dbl__inc__def,axiom,
    ! [Xa: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xa) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(Xa),Xa)),one_one_int) ).

% dbl_inc_def
tff(fact_7602_real__of__int__div__aux,axiom,
    ! [Xa: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Xa)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Xa),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(Xa),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_7603_real__of__int__div2,axiom,
    ! [Nb: int,Xa: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xa))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xa)))) ).

% real_of_int_div2
tff(fact_7604_real__of__int__div3,axiom,
    ! [Nb: int,Xa: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xa))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xa)))),one_one_real) ).

% real_of_int_div3
tff(fact_7605_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_7606_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_7607_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gp(L)),Qr) ).

% divmod_step_nat_def
tff(fact_7608_arctan__add,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)))) ) ) ) ).

% arctan_add
tff(fact_7609_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_gq(L),Qr) ).

% divmod_step_int_def
tff(fact_7610_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gp(L)),Qr) ).

% divmod_step_def
tff(fact_7611_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_gq(L),Qr) ).

% divmod_step_def
tff(fact_7612_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_gr(L),Qr) ).

% divmod_step_def
tff(fact_7613_floor__exists,axiom,
    ! [Xa: real] :
    ? [Z4: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z4)),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Z4),one_one_int))) ) ).

% floor_exists
tff(fact_7614_floor__exists,axiom,
    ! [Xa: rat] :
    ? [Z4: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z4)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Z4),one_one_int))) ) ).

% floor_exists
tff(fact_7615_floor__exists1,axiom,
    ! [Xa: real] :
    ? [X3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),Xa)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(X3),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Y4),one_one_int))) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
tff(fact_7616_floor__exists1,axiom,
    ! [Xa: rat] :
    ? [X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),Xa)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(X3),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),Xa)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Y4),one_one_int))) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
tff(fact_7617_divmod__nat__if,axiom,
    ! [Ma: nat,Nb: nat] :
      divmod_nat(Ma,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Ma),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gs),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_7618_divmod__BitM__2__eq,axiom,
    ! [Ma: num] : unique5052692396658037445od_int(aa_num_num(bitM,Ma),aa_num_num(bit0,one)) = product_Pair_int_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Ma)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_7619_set__decode__0,axiom,
    ! [Xa: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xa))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa) ) ).

% set_decode_0
tff(fact_7620_set__decode__Suc,axiom,
    ! [Nb: nat,Xa: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),aa_nat_set_nat(nat_set_decode,Xa))
    <=> aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_7621_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_7622_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_7623_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_7624_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_7625_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_7626_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_7627_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_7628_semiring__norm_I27_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(bitM,Nb)) ).

% semiring_norm(27)
tff(fact_7629_semiring__norm_I28_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).

% semiring_norm(28)
tff(fact_7630_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).

% eval_nat_numeral(2)
tff(fact_7631_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_7632_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_7633_numeral__BitM,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bitM,Nb)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,Nb))),one_one_complex) ).

% numeral_BitM
tff(fact_7634_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bitM,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb))),one_one_real) ).

% numeral_BitM
tff(fact_7635_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,Nb)) = aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),one_one_int) ).

% numeral_BitM
tff(fact_7636_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,Nb)) = aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb))),one_one_rat) ).

% numeral_BitM
tff(fact_7637_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_7638_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_7639_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_7640_divmod__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : divmod_nat(Ma,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) ).

% divmod_nat_def
tff(fact_7641_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_12: nat] : aa_nat_o(P,X_12)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_7642_ex__of__int__less,axiom,
    ! [Xa: real] :
    ? [Z4: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z4)),Xa) ).

% ex_of_int_less
tff(fact_7643_ex__of__int__less,axiom,
    ! [Xa: rat] :
    ? [Z4: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z4)),Xa) ).

% ex_of_int_less
tff(fact_7644_ex__less__of__int,axiom,
    ! [Xa: real] :
    ? [Z4: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_int_real(ring_1_of_int_real,Z4)) ).

% ex_less_of_int
tff(fact_7645_ex__less__of__int,axiom,
    ! [Xa: rat] :
    ? [Z4: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_int_rat(ring_1_of_int_rat,Z4)) ).

% ex_less_of_int
tff(fact_7646_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Z)) = insert_nat(Nb,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_7647_set__decode__def,axiom,
    ! [Xa: nat] : aa_nat_set_nat(nat_set_decode,Xa) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gt,Xa)) ).

% set_decode_def
tff(fact_7648_round__unique,axiom,
    ! [Xa: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xa) = Y ) ) ) ).

% round_unique
tff(fact_7649_round__unique,axiom,
    ! [Xa: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xa),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(plus_plus_rat(Xa),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,Xa) = Y ) ) ) ).

% round_unique
tff(fact_7650_round__unique_H,axiom,
    ! [Xa: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xa) = Nb ) ) ).

% round_unique'
tff(fact_7651_round__unique_H,axiom,
    ! [Xa: rat,Nb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xa),aa_int_rat(ring_1_of_int_rat,Nb)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,Xa) = Nb ) ) ).

% round_unique'
tff(fact_7652_of__int__round__abs__le,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xa))),Xa))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7653_of__int__round__abs__le,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))),Xa))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7654_of__int__round__gt,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xa))) ).

% of_int_round_gt
tff(fact_7655_of__int__round__gt,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xa),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))) ).

% of_int_round_gt
tff(fact_7656_of__int__round__ge,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xa))) ).

% of_int_round_ge
tff(fact_7657_of__int__round__ge,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Xa),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))) ).

% of_int_round_ge
tff(fact_7658_of__int__round__le,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_7659_of__int__round__le,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xa))),aa_rat_rat(plus_plus_rat(Xa),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_7660_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_7661_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_7662_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_7663_round__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_7664_round__neg__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_7665_round__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_7666_Sum__Icc__int,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ma),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_cr,set_or1266510415728281911st_int(Ma,Nb)) = aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(aa_int_int(times_times_int(Nb),aa_int_int(plus_plus_int(Nb),one_one_int))),aa_int_int(times_times_int(Ma),aa_int_int(minus_minus_int(Ma),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_7667_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_7668_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_7669_take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_se1745604003318907178nteger(Nb,A) = $ite(Nb = zero_zero_nat,zero_z3403309356797280102nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_7670_take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_7671_take__bit__rec,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_7672_tanh__real__altdef,axiom,
    ! [Xa: real] : aa_real_real(tanh_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)))) ).

% tanh_real_altdef
tff(fact_7673_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K),L) = $ite(
        ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K,aa_int_int(plus_plus_int(aa_int_int(ord_max_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_7674_or_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se1409905431419307370or_int(A),B)),B) = aa_int_int(bit_se1409905431419307370or_int(A),B) ).

% or.right_idem
tff(fact_7675_or_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ).

% or.right_idem
tff(fact_7676_or_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se1409905431419307370or_int(A),aa_int_int(bit_se1409905431419307370or_int(A),B)) = aa_int_int(bit_se1409905431419307370or_int(A),B) ).

% or.left_idem
tff(fact_7677_or_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ).

% or.left_idem
tff(fact_7678_or_Oidem,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),A) = A ).

% or.idem
tff(fact_7679_or_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),A) = A ).

% or.idem
tff(fact_7680_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_7681_take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_7682_take__bit__of__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_7683_or_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),zero_zero_int) = A ).

% or.right_neutral
tff(fact_7684_or_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),zero_zero_nat) = A ).

% or.right_neutral
tff(fact_7685_or_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(zero_zero_int),A) = A ).

% or.left_neutral
tff(fact_7686_or_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),A) = A ).

% or.left_neutral
tff(fact_7687_take__bit__and,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se725231765392027082nd_int(A),B)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) ).

% take_bit_and
tff(fact_7688_take__bit__and,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B)) ).

% take_bit_and
tff(fact_7689_take__bit__or,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se1409905431419307370or_int(A),B)) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) ).

% take_bit_or
tff(fact_7690_take__bit__or,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B)) ).

% take_bit_or
tff(fact_7691_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_concat_bit(Nb,K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(Nb),K) ).

% concat_bit_of_zero_2
tff(fact_7692_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_cm,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_7693_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_ci,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_7694_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cl),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_7695_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : groups6591440286371151544t_real(aTP_Lamm_gu,A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_7696_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_7697_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_7698_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_7699_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_7700_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_7701_sum_Oempty,axiom,
    ! [G: fun_o_complex] : groups5328290441151304332omplex(G,bot_bot_set_o) = zero_zero_complex ).

% sum.empty
tff(fact_7702_sum_Oempty,axiom,
    ! [G: fun_o_real] : groups8691415230153176458o_real(G,bot_bot_set_o) = zero_zero_real ).

% sum.empty
tff(fact_7703_sum_Oempty,axiom,
    ! [G: fun_o_rat] : groups7872700643590313910_o_rat(G,bot_bot_set_o) = zero_zero_rat ).

% sum.empty
tff(fact_7704_sum_Oempty,axiom,
    ! [G: fun_o_nat] : groups8507830703676809646_o_nat(G,bot_bot_set_o) = zero_zero_nat ).

% sum.empty
tff(fact_7705_sum_Oempty,axiom,
    ! [G: fun_o_int] : groups8505340233167759370_o_int(G,bot_bot_set_o) = zero_zero_int ).

% sum.empty
tff(fact_7706_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2073611262835488442omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_7707_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups3049146728041665814omplex(G,A2) = zero_zero_complex ) ) ).

% sum.infinite
tff(fact_7708_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_7709_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_7710_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7711_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7712_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_7713_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_7714_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_7715_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups3539618377306564664at_int(G,A2) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_7716_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),F3)
           => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7717_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),F3)
           => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7718_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),F3)
           => ( aa_nat_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_7719_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_7720_take__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).

% take_bit_0
tff(fact_7721_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_7722_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_7723_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_7724_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_7725_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_7726_take__bit__numeral__1,axiom,
    ! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),one_one_nat) = one_one_nat ).

% take_bit_numeral_1
tff(fact_7727_bit_Odisj__one__left,axiom,
    ! [Xa: code_integer] : bit_se1080825931792720795nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Xa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_left
tff(fact_7728_bit_Odisj__one__left,axiom,
    ! [Xa: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,one_one_int)),Xa) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_left
tff(fact_7729_bit_Odisj__one__right,axiom,
    ! [Xa: code_integer] : bit_se1080825931792720795nteger(Xa,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_right
tff(fact_7730_bit_Odisj__one__right,axiom,
    ! [Xa: int] : aa_int_int(bit_se1409905431419307370or_int(Xa),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_right
tff(fact_7731_mask__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = zero_zero_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_7732_mask__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,Nb) = zero_zero_int )
    <=> ( Nb = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_7733_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_7734_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_7735_exp__eq__one__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(exp_real,Xa) = one_one_real )
    <=> ( Xa = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_7736_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_7737_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_7738_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_gv(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_7739_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gw(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_7740_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_gx(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_7741_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_gy(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7742_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_gz(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7743_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ha(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_7744_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hb(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7745_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hc(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7746_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_hd(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7747_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_he(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_7748_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_hf(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_7749_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_hg(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_7750_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_hh(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_7751_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_hi(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7752_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_hj(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7753_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_hk(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_7754_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hl(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7755_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hm(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7756_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_hn(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7757_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ho(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_7758_sum_Oinsert,axiom,
    ! [A2: set_o,Xa: $o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((Xa)),A2)
       => ( groups8691415230153176458o_real(G,insert_o((Xa),A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_o_real(G,(Xa))),groups8691415230153176458o_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7759_sum_Oinsert,axiom,
    ! [A2: set_real,Xa: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xa),A2)
       => ( groups8097168146408367636l_real(G,insert_real(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xa)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7760_sum_Oinsert,axiom,
    ! [A2: set_int,Xa: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xa),A2)
       => ( groups8778361861064173332t_real(G,insert_int(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xa)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7761_sum_Oinsert,axiom,
    ! [A2: set_complex,Xa: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xa),A2)
       => ( groups5808333547571424918x_real(G,insert_complex(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xa)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7762_sum_Oinsert,axiom,
    ! [A2: set_o,Xa: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((Xa)),A2)
       => ( groups7872700643590313910_o_rat(G,insert_o((Xa),A2)) = aa_rat_rat(plus_plus_rat(aa_o_rat(G,(Xa))),groups7872700643590313910_o_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7763_sum_Oinsert,axiom,
    ! [A2: set_real,Xa: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xa),A2)
       => ( groups1300246762558778688al_rat(G,insert_real(Xa,A2)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xa)),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7764_sum_Oinsert,axiom,
    ! [A2: set_nat,Xa: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xa),A2)
       => ( groups2906978787729119204at_rat(G,insert_nat(Xa,A2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xa)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7765_sum_Oinsert,axiom,
    ! [A2: set_int,Xa: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xa),A2)
       => ( groups3906332499630173760nt_rat(G,insert_int(Xa,A2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xa)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7766_sum_Oinsert,axiom,
    ! [A2: set_complex,Xa: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xa),A2)
       => ( groups5058264527183730370ex_rat(G,insert_complex(Xa,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xa)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7767_sum_Oinsert,axiom,
    ! [A2: set_o,Xa: $o,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ~ aa_set_o_o(member_o((Xa)),A2)
       => ( groups8507830703676809646_o_nat(G,insert_o((Xa),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(G,(Xa))),groups8507830703676809646_o_nat(G,A2)) ) ) ) ).

% sum.insert
tff(fact_7768_take__bit__of__1__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = zero_zero_int )
    <=> ( Nb = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_7769_take__bit__of__1__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = zero_zero_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_7770_or__numerals_I8_J,axiom,
    ! [Xa: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa)) ).

% or_numerals(8)
tff(fact_7771_or__numerals_I8_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_numerals(8)
tff(fact_7772_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).

% or_numerals(2)
tff(fact_7773_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_numerals(2)
tff(fact_7774_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_7775_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_7776_one__less__exp__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% one_less_exp_iff
tff(fact_7777_exp__less__one__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_7778_exp__le__one__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_7779_one__le__exp__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% one_le_exp_iff
tff(fact_7780_take__bit__minus__one__eq__mask,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb) ).

% take_bit_minus_one_eq_mask
tff(fact_7781_take__bit__minus__one__eq__mask,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% take_bit_minus_one_eq_mask
tff(fact_7782_exp__ln__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xa)) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% exp_ln_iff
tff(fact_7783_exp__ln,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xa)) = Xa ) ) ).

% exp_ln
tff(fact_7784_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_7785_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_hp(F),A2)) ).

% sum_abs_ge_zero
tff(fact_7786_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_hq(F),A2)) ).

% sum_abs_ge_zero
tff(fact_7787_or__numerals_I3_J,axiom,
    ! [Xa: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y))) ).

% or_numerals(3)
tff(fact_7788_or__numerals_I3_J,axiom,
    ! [Xa: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),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_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y))) ).

% or_numerals(3)
tff(fact_7789_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y)) ).

% or_numerals(1)
tff(fact_7790_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_numerals(1)
tff(fact_7791_or__numerals_I5_J,axiom,
    ! [Xa: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa)) ).

% or_numerals(5)
tff(fact_7792_or__numerals_I5_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_numerals(5)
tff(fact_7793_take__bit__of__1,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,one_one_Code_integer) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_7794_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_7795_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_7796_or__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_7797_or__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_7798_sum__of__bool__mult__eq,axiom,
    ! [A2: set_real,P: fun_real_o,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_hr(P),F),A2) = groups8097168146408367636l_real(F,inf_inf_set_real(A2,collect_real(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7799_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: fun_int_o,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_hs(P),F),A2) = groups8778361861064173332t_real(F,inf_inf_set_int(A2,collect_int(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7800_sum__of__bool__mult__eq,axiom,
    ! [A2: set_complex,P: fun_complex_o,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ht(P),F),A2) = groups5808333547571424918x_real(F,inf_inf_set_complex(A2,collect_complex(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7801_sum__of__bool__mult__eq,axiom,
    ! [A2: set_real,P: fun_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hu(P),F),A2) = groups1300246762558778688al_rat(F,inf_inf_set_real(A2,collect_real(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7802_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: fun_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_hv(P),F),A2) = groups3906332499630173760nt_rat(F,inf_inf_set_int(A2,collect_int(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7803_sum__of__bool__mult__eq,axiom,
    ! [A2: set_complex,P: fun_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_hw(P),F),A2) = groups5058264527183730370ex_rat(F,inf_inf_set_complex(A2,collect_complex(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7804_sum__of__bool__mult__eq,axiom,
    ! [A2: set_nat,P: fun_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hx(P),F),A2) = groups2906978787729119204at_rat(F,inf_inf_set_nat(A2,collect_nat(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7805_sum__of__bool__mult__eq,axiom,
    ! [A2: set_real,P: fun_real_o,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_hy(P),F),A2) = groups1935376822645274424al_nat(F,inf_inf_set_real(A2,collect_real(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7806_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: fun_int_o,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_hz(P),F),A2) = groups4541462559716669496nt_nat(F,inf_inf_set_int(A2,collect_int(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7807_sum__of__bool__mult__eq,axiom,
    ! [A2: set_complex,P: fun_complex_o,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ia(P),F),A2) = groups5693394587270226106ex_nat(F,inf_inf_set_complex(A2,collect_complex(P))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_7808_sum__mult__of__bool__eq,axiom,
    ! [A2: set_real,F: fun_real_real,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_ib(F),P),A2) = groups8097168146408367636l_real(F,inf_inf_set_real(A2,collect_real(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7809_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_ic(F),P),A2) = groups8778361861064173332t_real(F,inf_inf_set_int(A2,collect_int(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7810_sum__mult__of__bool__eq,axiom,
    ! [A2: set_complex,F: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_id(F),P),A2) = groups5808333547571424918x_real(F,inf_inf_set_complex(A2,collect_complex(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7811_sum__mult__of__bool__eq,axiom,
    ! [A2: set_real,F: fun_real_rat,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_ie(F),P),A2) = groups1300246762558778688al_rat(F,inf_inf_set_real(A2,collect_real(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7812_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_if(F),P),A2) = groups3906332499630173760nt_rat(F,inf_inf_set_int(A2,collect_int(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7813_sum__mult__of__bool__eq,axiom,
    ! [A2: set_complex,F: fun_complex_rat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_ig(F),P),A2) = groups5058264527183730370ex_rat(F,inf_inf_set_complex(A2,collect_complex(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7814_sum__mult__of__bool__eq,axiom,
    ! [A2: set_nat,F: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ih(F),P),A2) = groups2906978787729119204at_rat(F,inf_inf_set_nat(A2,collect_nat(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7815_sum__mult__of__bool__eq,axiom,
    ! [A2: set_real,F: fun_real_nat,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(aa_fun5513133441921015042al_nat(aTP_Lamm_ii(F),P),A2) = groups1935376822645274424al_nat(F,inf_inf_set_real(A2,collect_real(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7816_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_ij(F),P),A2) = groups4541462559716669496nt_nat(F,inf_inf_set_int(A2,collect_int(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7817_sum__mult__of__bool__eq,axiom,
    ! [A2: set_complex,F: fun_complex_nat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(aa_fun7518244829075205762ex_nat(aTP_Lamm_ik(F),P),A2) = groups5693394587270226106ex_nat(F,inf_inf_set_complex(A2,collect_complex(P))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_7818_even__take__bit__eq,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(Nb,A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_7819_even__take__bit__eq,axiom,
    ! [Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(Nb),A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_7820_even__take__bit__eq,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_7821_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_7822_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_7823_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_7824_or__numerals_I4_J,axiom,
    ! [Xa: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(4)
tff(fact_7825_or__numerals_I4_J,axiom,
    ! [Xa: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),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_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(4)
tff(fact_7826_or__numerals_I6_J,axiom,
    ! [Xa: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(6)
tff(fact_7827_or__numerals_I6_J,axiom,
    ! [Xa: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(6)
tff(fact_7828_or__numerals_I7_J,axiom,
    ! [Xa: num,Y: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xa))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Xa)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(7)
tff(fact_7829_or__numerals_I7_J,axiom,
    ! [Xa: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),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_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xa)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(7)
tff(fact_7830_take__bit__of__exp,axiom,
    ! [Ma: nat,Nb: nat] : bit_se1745604003318907178nteger(Ma,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_7831_take__bit__of__exp,axiom,
    ! [Ma: nat,Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Ma),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_7832_take__bit__of__exp,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_7833_take__bit__of__2,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_7834_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_7835_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_7836_take__bit__eq__mask,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se725231765392027082nd_int(A),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% take_bit_eq_mask
tff(fact_7837_take__bit__eq__mask,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% take_bit_eq_mask
tff(fact_7838_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se1409905431419307370or_int(A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_7839_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_7840_bit_Odisj__zero__right,axiom,
    ! [Xa: int] : aa_int_int(bit_se1409905431419307370or_int(Xa),zero_zero_int) = Xa ).

% bit.disj_zero_right
tff(fact_7841_or_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(bit_se1409905431419307370or_int(B),aa_int_int(bit_se1409905431419307370or_int(A),C)) = aa_int_int(bit_se1409905431419307370or_int(A),aa_int_int(bit_se1409905431419307370or_int(B),C)) ).

% or.left_commute
tff(fact_7842_or_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),C)) ).

% or.left_commute
tff(fact_7843_or_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se1409905431419307370or_int(A),B) = aa_int_int(bit_se1409905431419307370or_int(B),A) ).

% or.commute
tff(fact_7844_or_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),A) ).

% or.commute
tff(fact_7845_or_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se1409905431419307370or_int(A),B)),C) = aa_int_int(bit_se1409905431419307370or_int(A),aa_int_int(bit_se1409905431419307370or_int(B),C)) ).

% or.assoc
tff(fact_7846_or_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),C)) ).

% or.assoc
tff(fact_7847_sum_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => ( aa_int_int(G,X3) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A2) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_7848_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => ( aa_complex_complex(G,X3) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_7849_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => ( aa_nat_nat(G,X3) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_7850_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => ( aa_nat_real(G,X3) = zero_zero_real ) )
     => ( groups6591440286371151544t_real(G,A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_7851_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups5754745047067104278omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: real] :
            ( aa_set_real_o(member_real(A4),A2)
           => ( aa_real_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7852_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( groups2073611262835488442omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A2)
           => ( aa_nat_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7853_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups3049146728041665814omplex(G,A2) != zero_zero_complex )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7854_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups5808333547571424918x_real(G,A2) != zero_zero_real )
     => ~ ! [A4: complex] :
            ( aa_set_complex_o(member_complex(A4),A2)
           => ( aa_complex_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7855_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A4: real] :
            ( aa_set_real_o(member_real(A4),A2)
           => ( aa_real_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7856_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7857_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A2: set_complex] :
      ( ( groups5058264527183730370ex_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: complex] :
            ( aa_set_complex_o(member_complex(A4),A2)
           => ( aa_complex_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7858_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups1300246762558778688al_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: real] :
            ( aa_set_real_o(member_real(A4),A2)
           => ( aa_real_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7859_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A2)
           => ( aa_nat_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7860_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A2) != zero_zero_rat )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7861_take__bit__add,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(A),B)) ).

% take_bit_add
tff(fact_7862_take__bit__add,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_7863_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma)),Ma) ).

% take_bit_nat_less_eq_self
tff(fact_7864_take__bit__tightened__less__eq__nat,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_7865_take__bit__tightened,axiom,
    ! [Nb: nat,A: int,B: int,Ma: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_int_int(bit_se2923211474154528505it_int(Ma),A) = aa_int_int(bit_se2923211474154528505it_int(Ma),B) ) ) ) ).

% take_bit_tightened
tff(fact_7866_take__bit__tightened,axiom,
    ! [Nb: nat,A: nat,B: nat,Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),B) ) ) ) ).

% take_bit_tightened
tff(fact_7867_take__bit__of__int,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_of_int
tff(fact_7868_of__int__mask__eq,axiom,
    ! [Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% of_int_mask_eq
tff(fact_7869_of__int__or__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se1409905431419307370or_int(K),L)) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_or_eq
tff(fact_7870_exp__not__eq__zero,axiom,
    ! [Xa: complex] : aa_complex_complex(exp_complex,Xa) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_7871_exp__not__eq__zero,axiom,
    ! [Xa: real] : aa_real_real(exp_real,Xa) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_7872_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_7873_exp__times__arg__commute,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A2)),A2) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(exp_complex,A2)) ).

% exp_times_arg_commute
tff(fact_7874_exp__times__arg__commute,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,A2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(exp_real,A2)) ).

% exp_times_arg_commute
tff(fact_7875_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(times_times_int(K),L)) ).

% take_bit_mult
tff(fact_7876_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(K),L)) ).

% take_bit_diff
tff(fact_7877_bit_Odisj__conj__distrib2,axiom,
    ! [Y: int,Z: int,Xa: int] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se725231765392027082nd_int(Y),Z)),Xa) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se1409905431419307370or_int(Y),Xa)),aa_int_int(bit_se1409905431419307370or_int(Z),Xa)) ).

% bit.disj_conj_distrib2
tff(fact_7878_bit_Oconj__disj__distrib2,axiom,
    ! [Y: int,Z: int,Xa: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se1409905431419307370or_int(Y),Z)),Xa) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se725231765392027082nd_int(Y),Xa)),aa_int_int(bit_se725231765392027082nd_int(Z),Xa)) ).

% bit.conj_disj_distrib2
tff(fact_7879_bit_Odisj__conj__distrib,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(bit_se1409905431419307370or_int(Xa),aa_int_int(bit_se725231765392027082nd_int(Y),Z)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se1409905431419307370or_int(Xa),Y)),aa_int_int(bit_se1409905431419307370or_int(Xa),Z)) ).

% bit.disj_conj_distrib
tff(fact_7880_bit_Oconj__disj__distrib,axiom,
    ! [Xa: int,Y: int,Z: int] : aa_int_int(bit_se725231765392027082nd_int(Xa),aa_int_int(bit_se1409905431419307370or_int(Y),Z)) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se725231765392027082nd_int(Xa),Y)),aa_int_int(bit_se725231765392027082nd_int(Xa),Z)) ).

% bit.conj_disj_distrib
tff(fact_7881_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B: int] : bit_concat_bit(Nb,aa_int_int(bit_se2923211474154528505it_int(Nb),B)) = bit_concat_bit(Nb,B) ).

% concat_bit_take_bit_eq
tff(fact_7882_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R2: int,Sb: int] :
      ( ( aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(bit_concat_bit(Nb,R2),Sb) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(bit_se2923211474154528505it_int(Nb),R2) )
        & ( L = Sb ) ) ) ).

% concat_bit_eq_iff
tff(fact_7883_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_7884_sum__product,axiom,
    ! [F: fun_int_int,A2: set_int,G: fun_int_int,B2: set_int] : aa_int_int(times_times_int(groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,B2)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_im(F),G),B2),A2) ).

% sum_product
tff(fact_7885_sum__product,axiom,
    ! [F: fun_complex_complex,A2: set_complex,G: fun_complex_complex,B2: set_complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,B2)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_io(F),G),B2),A2) ).

% sum_product
tff(fact_7886_sum__product,axiom,
    ! [F: fun_nat_nat,A2: set_nat,G: fun_nat_nat,B2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_iq(F),G),B2)),A2) ).

% sum_product
tff(fact_7887_sum__product,axiom,
    ! [F: fun_nat_real,A2: set_nat,G: fun_nat_real,B2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,B2)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_is(F),G),B2),A2) ).

% sum_product
tff(fact_7888_sum__distrib__right,axiom,
    ! [F: fun_int_int,A2: set_int,R2: int] : aa_int_int(times_times_int(groups4538972089207619220nt_int(F,A2)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_it(F),R2),A2) ).

% sum_distrib_right
tff(fact_7889_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_iu(F),R2),A2) ).

% sum_distrib_right
tff(fact_7890_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A2: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_iv(F),R2)),A2) ).

% sum_distrib_right
tff(fact_7891_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iw(F),R2),A2) ).

% sum_distrib_right
tff(fact_7892_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A2: set_int] : aa_int_int(times_times_int(R2),groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ix(R2),F),A2) ).

% sum_distrib_left
tff(fact_7893_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A2: set_complex] : aa_complex_complex(times_times_complex(R2),groups7754918857620584856omplex(F,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_iy(R2),F),A2) ).

% sum_distrib_left
tff(fact_7894_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_iz(R2),F)),A2) ).

% sum_distrib_left
tff(fact_7895_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),groups6591440286371151544t_real(F,A2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ja(R2),F),A2) ).

% sum_distrib_left
tff(fact_7896_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_jb(G),H),A2) = aa_int_int(plus_plus_int(groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).

% sum.distrib
tff(fact_7897_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_jc(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_7898_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jd(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_7899_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_je(G),H),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A2)),groups6591440286371151544t_real(H,A2)) ).

% sum.distrib
tff(fact_7900_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(divide1717551699836669952omplex(groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_jf(F),R2),A2) ).

% sum_divide_distrib
tff(fact_7901_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups6591440286371151544t_real(F,A2)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jg(F),R2),A2) ).

% sum_divide_distrib
tff(fact_7902_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_jh(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(F,A2)),A) ).

% mod_sum_eq
tff(fact_7903_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ).

% mod_sum_eq
tff(fact_7904_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_7905_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A2)) ) ).

% sum_nonneg
tff(fact_7906_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2)) ) ).

% sum_nonneg
tff(fact_7907_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2)) ) ).

% sum_nonneg
tff(fact_7908_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7909_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7910_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7911_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_7912_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_7913_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_7914_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_7915_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7916_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7917_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7918_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7919_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7920_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7921_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7922_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7923_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7924_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7925_not__exp__less__zero,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_7926_exp__gt__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xa)) ).

% exp_gt_zero
tff(fact_7927_exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X3: real] : aa_real_real(exp_real,X3) = Y ) ).

% exp_total
tff(fact_7928_not__exp__le__zero,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_7929_exp__ge__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xa)) ).

% exp_ge_zero
tff(fact_7930_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se1409905431419307370or_int(K),L)) ) ).

% or_greater_eq
tff(fact_7931_OR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(Xa),Y)) ) ) ).

% OR_lower
tff(fact_7932_take__bit__tightened__less__eq__int,axiom,
    ! [Ma: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Ma),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_7933_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_ri631733984087533419it_int(Nb),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_7934_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_7935_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_nonnegative
tff(fact_7936_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_7937_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_7938_signed__take__bit__take__bit,axiom,
    ! [Ma: nat,Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),bit_se2923211474154528505it_int(Nb),bit_ri631733984087533419it_int(Ma)),A) ).

% signed_take_bit_take_bit
tff(fact_7939_mult__exp__exp,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xa)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xa),Y)) ).

% mult_exp_exp
tff(fact_7940_mult__exp__exp,axiom,
    ! [Xa: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) ).

% mult_exp_exp
tff(fact_7941_exp__add__commuting,axiom,
    ! [Xa: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(Xa),Y) = aa_complex_complex(times_times_complex(Y),Xa) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xa),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xa)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_7942_exp__add__commuting,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xa) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_7943_exp__diff,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(Xa),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,Xa)),aa_complex_complex(exp_complex,Y)) ).

% exp_diff
tff(fact_7944_exp__diff,axiom,
    ! [Xa: real,Y: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y)) ).

% exp_diff
tff(fact_7945_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se4203085406695923979it_int(Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se4203085406695923979it_int(Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_7946_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_7947_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se7879613467334960850it_int(Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se7879613467334960850it_int(Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_7948_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_7949_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se2159334234014336723it_int(Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se2159334234014336723it_int(Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_7950_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_7951_plus__and__or,axiom,
    ! [Xa: int,Y: int] : aa_int_int(plus_plus_int(aa_int_int(bit_se725231765392027082nd_int(Xa),Y)),aa_int_int(bit_se1409905431419307370or_int(Xa),Y)) = aa_int_int(plus_plus_int(Xa),Y) ).

% plus_and_or
tff(fact_7952_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ay(A2),P))) = groups5754745047067104278omplex(aa_fun1098856492951973088omplex(aTP_Lamm_jj(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7953_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2073611262835488442omplex(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bc(A2),P))) = groups2073611262835488442omplex(aa_fun1400689748890728928omplex(aTP_Lamm_jk(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7954_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bb(A2),P))) = groups3049146728041665814omplex(aa_fun6439487526662521568omplex(aTP_Lamm_jl(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7955_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ay(A2),P))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_jm(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7956_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bb(A2),P))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_jn(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7957_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ax(A2),P))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_jo(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7958_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ay(A2),P))) = groups1300246762558778688al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_jp(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7959_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bc(A2),P))) = groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_jq(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7960_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bb(A2),P))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_jr(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7961_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_rat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ax(A2),P))) = groups5058264527183730370ex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_js(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_7962_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_nonnegative_int
tff(fact_7963_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_7964_mask__Suc__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% mask_Suc_exp
tff(fact_7965_mask__Suc__exp,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_Suc_exp
tff(fact_7966_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
             => ( aa_real_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7967_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
             => ( aa_int_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7968_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A2)
             => ( aa_complex_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7969_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
             => ( aa_real_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7970_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
       => ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A2)
             => ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7971_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
       => ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
             => ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7972_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
       => ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A2)
             => ( aa_complex_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7973_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
       => ( ( groups1935376822645274424al_nat(F,A2) = zero_zero_nat )
        <=> ! [X4: real] :
              ( aa_set_real_o(member_real(X4),A2)
             => ( aa_real_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7974_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
       => ( ( groups4541462559716669496nt_nat(F,A2) = zero_zero_nat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A2)
             => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7975_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
       => ( ( groups5693394587270226106ex_nat(F,A2) = zero_zero_nat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A2)
             => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7976_sum__le__included,axiom,
    ! [Sb: set_int,Ta: set_int,G: fun_int_real,Ia: fun_int_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),Sb)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),Ta)
                    & ( aa_int_int(Ia,Xa2) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_int_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,Sb)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7977_sum__le__included,axiom,
    ! [Sb: set_int,Ta: set_complex,G: fun_complex_real,Ia: fun_complex_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),Sb)
               => ? [Xa2: complex] :
                    ( aa_set_complex_o(member_complex(Xa2),Ta)
                    & ( aa_complex_int(Ia,Xa2) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_complex_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,Sb)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7978_sum__le__included,axiom,
    ! [Sb: set_complex,Ta: set_int,G: fun_int_real,Ia: fun_int_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X3)) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),Sb)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),Ta)
                    & ( aa_int_complex(Ia,Xa2) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_int_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,Sb)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7979_sum__le__included,axiom,
    ! [Sb: set_complex,Ta: set_complex,G: fun_complex_real,Ia: fun_complex_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X3)) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),Sb)
               => ? [Xa2: complex] :
                    ( aa_set_complex_o(member_complex(Xa2),Ta)
                    & ( aa_complex_complex(Ia,Xa2) = X3 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_complex_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,Sb)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7980_sum__le__included,axiom,
    ! [Sb: set_nat,Ta: set_nat,G: fun_nat_rat,Ia: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),Sb)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),Ta)
                    & ( aa_nat_nat(Ia,Xa2) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,Xa2)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,Sb)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7981_sum__le__included,axiom,
    ! [Sb: set_nat,Ta: set_int,G: fun_int_rat,Ia: fun_int_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),Sb)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),Ta)
                    & ( aa_int_nat(Ia,Xa2) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_int_rat(G,Xa2)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,Sb)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7982_sum__le__included,axiom,
    ! [Sb: set_nat,Ta: set_complex,G: fun_complex_rat,Ia: fun_complex_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),Sb)
               => ? [Xa2: complex] :
                    ( aa_set_complex_o(member_complex(Xa2),Ta)
                    & ( aa_complex_nat(Ia,Xa2) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_complex_rat(G,Xa2)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,Sb)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7983_sum__le__included,axiom,
    ! [Sb: set_int,Ta: set_nat,G: fun_nat_rat,Ia: fun_nat_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),Sb)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),Ta)
                    & ( aa_nat_int(Ia,Xa2) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_nat_rat(G,Xa2)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,Sb)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7984_sum__le__included,axiom,
    ! [Sb: set_int,Ta: set_int,G: fun_int_rat,Ia: fun_int_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),Sb)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),Ta)
                    & ( aa_int_int(Ia,Xa2) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,Xa2)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,Sb)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7985_sum__le__included,axiom,
    ! [Sb: set_int,Ta: set_complex,G: fun_complex_rat,Ia: fun_complex_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),Sb)
               => ? [Xa2: complex] :
                    ( aa_set_complex_o(member_complex(Xa2),Ta)
                    & ( aa_complex_int(Ia,Xa2) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_complex_rat(G,Xa2)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,Sb)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7986_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7987_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7988_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7989_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7990_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7991_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7992_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7993_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X)),aa_nat_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7994_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X3)),aa_complex_int(G,X3)) )
       => ( ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X)),aa_complex_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7995_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(G,X3)) )
       => ( ? [X: int] :
              ( aa_set_int_o(member_int(X),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X)),aa_int_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7996_sum_Orelated,axiom,
    ! [R3: fun_co1148497455491383662plex_o,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R3,zero_zero_complex),zero_zero_complex)
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R3,X1),X23)
              & aa_complex_o(aa_com5808801942499186519plex_o(R3,Y1),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R3,aa_complex_complex(plus_plus_complex(X1),Y1)),aa_complex_complex(plus_plus_complex(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R3,aa_nat_complex(H,X3)),aa_nat_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R3,groups2073611262835488442omplex(H,S2)),groups2073611262835488442omplex(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7997_sum_Orelated,axiom,
    ! [R3: fun_co1148497455491383662plex_o,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R3,zero_zero_complex),zero_zero_complex)
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R3,X1),X23)
              & aa_complex_o(aa_com5808801942499186519plex_o(R3,Y1),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R3,aa_complex_complex(plus_plus_complex(X1),Y1)),aa_complex_complex(plus_plus_complex(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R3,aa_int_complex(H,X3)),aa_int_complex(G,X3)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R3,groups3049146728041665814omplex(H,S2)),groups3049146728041665814omplex(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7998_sum_Orelated,axiom,
    ! [R3: fun_real_fun_real_o,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R3,zero_zero_real),zero_zero_real)
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R3,X1),X23)
              & aa_real_o(aa_real_fun_real_o(R3,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_real_o(aa_real_fun_real_o(R3,aa_int_real(H,X3)),aa_int_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R3,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7999_sum_Orelated,axiom,
    ! [R3: fun_real_fun_real_o,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R3,zero_zero_real),zero_zero_real)
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R3,X1),X23)
              & aa_real_o(aa_real_fun_real_o(R3,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_real_o(aa_real_fun_real_o(R3,aa_complex_real(H,X3)),aa_complex_real(G,X3)) )
           => aa_real_o(aa_real_fun_real_o(R3,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_8000_sum_Orelated,axiom,
    ! [R3: fun_rat_fun_rat_o,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R3,zero_zero_rat),zero_zero_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R3,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R3,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R3,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R3,aa_nat_rat(H,X3)),aa_nat_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R3,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_8001_sum_Orelated,axiom,
    ! [R3: fun_rat_fun_rat_o,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R3,zero_zero_rat),zero_zero_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R3,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R3,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R3,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R3,aa_int_rat(H,X3)),aa_int_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R3,groups3906332499630173760nt_rat(H,S2)),groups3906332499630173760nt_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_8002_sum_Orelated,axiom,
    ! [R3: fun_rat_fun_rat_o,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R3,zero_zero_rat),zero_zero_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R3,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R3,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R3,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R3,aa_complex_rat(H,X3)),aa_complex_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R3,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_8003_sum_Orelated,axiom,
    ! [R3: fun_nat_fun_nat_o,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R3,zero_zero_nat),zero_zero_nat)
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R3,X1),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R3,Y1),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R3,aa_int_nat(H,X3)),aa_int_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(R3,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_8004_sum_Orelated,axiom,
    ! [R3: fun_nat_fun_nat_o,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R3,zero_zero_nat),zero_zero_nat)
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R3,X1),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R3,Y1),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R3,aa_complex_nat(H,X3)),aa_complex_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(R3,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_8005_sum_Orelated,axiom,
    ! [R3: fun_int_fun_int_o,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R3,zero_zero_int),zero_zero_int)
     => ( ! [X1: int,Y1: int,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R3,X1),X23)
              & aa_int_o(aa_int_fun_int_o(R3,Y1),Y23) )
           => aa_int_o(aa_int_fun_int_o(R3,aa_int_int(plus_plus_int(X1),Y1)),aa_int_int(plus_plus_int(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => aa_int_o(aa_int_fun_int_o(R3,aa_nat_int(H,X3)),aa_nat_int(G,X3)) )
           => aa_int_o(aa_int_fun_int_o(R3,groups3539618377306564664at_int(H,S2)),groups3539618377306564664at_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_8006_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8007_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8008_sum__strict__mono,axiom,
    ! [A2: set_o,F: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(F,(X3))),aa_o_real(G,(X3))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8691415230153176458o_real(F,A2)),groups8691415230153176458o_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8009_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8010_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8011_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8012_sum__strict__mono,axiom,
    ! [A2: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(X3))),aa_o_rat(G,(X3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(F,A2)),groups7872700643590313910_o_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8013_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8014_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8015_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_8016_sum_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_real,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8691415230153176458o_real(G,insert_o((Xa),A2)) = $ite(aa_set_o_o(member_o((Xa)),A2),groups8691415230153176458o_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_o_real(G,(Xa))),groups8691415230153176458o_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8017_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,insert_real(Xa,A2)) = $ite(aa_set_real_o(member_real(Xa),A2),groups8097168146408367636l_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xa)),groups8097168146408367636l_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8018_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_real,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,insert_int(Xa,A2)) = $ite(aa_set_int_o(member_int(Xa),A2),groups8778361861064173332t_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xa)),groups8778361861064173332t_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8019_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_real,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,insert_complex(Xa,A2)) = $ite(aa_set_complex_o(member_complex(Xa),A2),groups5808333547571424918x_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xa)),groups5808333547571424918x_real(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8020_sum_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_rat,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups7872700643590313910_o_rat(G,insert_o((Xa),A2)) = $ite(aa_set_o_o(member_o((Xa)),A2),groups7872700643590313910_o_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_o_rat(G,(Xa))),groups7872700643590313910_o_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8021_sum_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_rat,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,insert_real(Xa,A2)) = $ite(aa_set_real_o(member_real(Xa),A2),groups1300246762558778688al_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xa)),groups1300246762558778688al_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8022_sum_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_rat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,insert_nat(Xa,A2)) = $ite(aa_set_nat_o(member_nat(Xa),A2),groups2906978787729119204at_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xa)),groups2906978787729119204at_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8023_sum_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_rat,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,insert_int(Xa,A2)) = $ite(aa_set_int_o(member_int(Xa),A2),groups3906332499630173760nt_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_int_rat(G,Xa)),groups3906332499630173760nt_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8024_sum_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_rat,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,insert_complex(Xa,A2)) = $ite(aa_set_complex_o(member_complex(Xa),A2),groups5058264527183730370ex_rat(G,A2),aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xa)),groups5058264527183730370ex_rat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8025_sum_Oinsert__if,axiom,
    ! [A2: set_o,G: fun_o_nat,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8507830703676809646_o_nat(G,insert_o((Xa),A2)) = $ite(aa_set_o_o(member_o((Xa)),A2),groups8507830703676809646_o_nat(G,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(G,(Xa))),groups8507830703676809646_o_nat(G,A2))) ) ) ).

% sum.insert_if
tff(fact_8026_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S2: set_real,Ia: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_real_o(finite_finite_real,T4)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
             => ( aa_real_real(Ia,aa_real_real(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
               => aa_set_real_o(member_real(aa_real_real(J,A4)),minus_minus_set_real(T2,T4)) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(T2,T4))
                 => ( aa_real_real(J,aa_real_real(Ia,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( aa_set_real_o(member_real(B4),minus_minus_set_real(T2,T4))
                   => aa_set_real_o(member_real(aa_real_real(Ia,B4)),minus_minus_set_real(S2,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( aa_set_real_o(member_real(B4),T4)
                       => ( aa_real_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S2)
                         => ( aa_real_complex(H,aa_real_real(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8027_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S2: set_real,Ia: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_int_o(finite_finite_int,T4)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
             => ( aa_int_real(Ia,aa_real_int(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
               => aa_set_int_o(member_int(aa_real_int(J,A4)),minus_minus_set_int(T2,T4)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T2,T4))
                 => ( aa_real_int(J,aa_int_real(Ia,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T2,T4))
                   => aa_set_real_o(member_real(aa_int_real(Ia,B4)),minus_minus_set_real(S2,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T4)
                       => ( aa_int_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S2)
                         => ( aa_int_complex(H,aa_real_int(J,A4)) = aa_real_complex(G,A4) ) )
                     => ( groups5754745047067104278omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8028_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S2: set_int,Ia: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_real_o(finite_finite_real,T4)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
             => ( aa_real_int(Ia,aa_int_real(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
               => aa_set_real_o(member_real(aa_int_real(J,A4)),minus_minus_set_real(T2,T4)) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(T2,T4))
                 => ( aa_int_real(J,aa_real_int(Ia,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( aa_set_real_o(member_real(B4),minus_minus_set_real(T2,T4))
                   => aa_set_int_o(member_int(aa_real_int(Ia,B4)),minus_minus_set_int(S2,S4)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S4)
                     => ( aa_int_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( aa_set_real_o(member_real(B4),T4)
                       => ( aa_real_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S2)
                         => ( aa_real_complex(H,aa_int_real(J,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8029_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S2: set_int,Ia: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_int_o(finite_finite_int,T4)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
             => ( aa_int_int(Ia,aa_int_int(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
               => aa_set_int_o(member_int(aa_int_int(J,A4)),minus_minus_set_int(T2,T4)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T2,T4))
                 => ( aa_int_int(J,aa_int_int(Ia,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T2,T4))
                   => aa_set_int_o(member_int(aa_int_int(Ia,B4)),minus_minus_set_int(S2,S4)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S4)
                     => ( aa_int_complex(G,A4) = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T4)
                       => ( aa_int_complex(H,B4) = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S2)
                         => ( aa_int_complex(H,aa_int_int(J,A4)) = aa_int_complex(G,A4) ) )
                     => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8030_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S2: set_real,Ia: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_real_o(finite_finite_real,T4)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
             => ( aa_real_real(Ia,aa_real_real(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
               => aa_set_real_o(member_real(aa_real_real(J,A4)),minus_minus_set_real(T2,T4)) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(T2,T4))
                 => ( aa_real_real(J,aa_real_real(Ia,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( aa_set_real_o(member_real(B4),minus_minus_set_real(T2,T4))
                   => aa_set_real_o(member_real(aa_real_real(Ia,B4)),minus_minus_set_real(S2,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_real(G,A4) = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( aa_set_real_o(member_real(B4),T4)
                       => ( aa_real_real(H,B4) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S2)
                         => ( aa_real_real(H,aa_real_real(J,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8031_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S2: set_real,Ia: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_real,H: fun_int_real] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_int_o(finite_finite_int,T4)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
             => ( aa_int_real(Ia,aa_real_int(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
               => aa_set_int_o(member_int(aa_real_int(J,A4)),minus_minus_set_int(T2,T4)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T2,T4))
                 => ( aa_real_int(J,aa_int_real(Ia,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T2,T4))
                   => aa_set_real_o(member_real(aa_int_real(Ia,B4)),minus_minus_set_real(S2,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_real(G,A4) = zero_zero_real ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T4)
                       => ( aa_int_real(H,B4) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S2)
                         => ( aa_int_real(H,aa_real_int(J,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8032_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_complex,S2: set_real,Ia: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T4)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
             => ( aa_complex_real(Ia,aa_real_complex(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S2,S4))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A4)),minus_811609699411566653omplex(T2,T4)) )
           => ( ! [B4: complex] :
                  ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T2,T4))
                 => ( aa_real_complex(J,aa_complex_real(Ia,B4)) = B4 ) )
             => ( ! [B4: complex] :
                    ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T2,T4))
                   => aa_set_real_o(member_real(aa_complex_real(Ia,B4)),minus_minus_set_real(S2,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_real(G,A4) = zero_zero_real ) )
                 => ( ! [B4: complex] :
                        ( aa_set_complex_o(member_complex(B4),T4)
                       => ( aa_complex_real(H,B4) = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S2)
                         => ( aa_complex_real(H,aa_real_complex(J,A4)) = aa_real_real(G,A4) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8033_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S2: set_int,Ia: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_real,H: fun_real_real] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_real_o(finite_finite_real,T4)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
             => ( aa_real_int(Ia,aa_int_real(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
               => aa_set_real_o(member_real(aa_int_real(J,A4)),minus_minus_set_real(T2,T4)) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(T2,T4))
                 => ( aa_int_real(J,aa_real_int(Ia,B4)) = B4 ) )
             => ( ! [B4: real] :
                    ( aa_set_real_o(member_real(B4),minus_minus_set_real(T2,T4))
                   => aa_set_int_o(member_int(aa_real_int(Ia,B4)),minus_minus_set_int(S2,S4)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S4)
                     => ( aa_int_real(G,A4) = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( aa_set_real_o(member_real(B4),T4)
                       => ( aa_real_real(H,B4) = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S2)
                         => ( aa_real_real(H,aa_int_real(J,A4)) = aa_int_real(G,A4) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8034_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S2: set_int,Ia: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_int_o(finite_finite_int,T4)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
             => ( aa_int_int(Ia,aa_int_int(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
               => aa_set_int_o(member_int(aa_int_int(J,A4)),minus_minus_set_int(T2,T4)) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(T2,T4))
                 => ( aa_int_int(J,aa_int_int(Ia,B4)) = B4 ) )
             => ( ! [B4: int] :
                    ( aa_set_int_o(member_int(B4),minus_minus_set_int(T2,T4))
                   => aa_set_int_o(member_int(aa_int_int(Ia,B4)),minus_minus_set_int(S2,S4)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S4)
                     => ( aa_int_real(G,A4) = zero_zero_real ) )
                 => ( ! [B4: int] :
                        ( aa_set_int_o(member_int(B4),T4)
                       => ( aa_int_real(H,B4) = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S2)
                         => ( aa_int_real(H,aa_int_int(J,A4)) = aa_int_real(G,A4) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8035_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_complex,S2: set_int,Ia: fun_complex_int,J: fun_int_complex,T2: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T4)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
             => ( aa_complex_int(Ia,aa_int_complex(J,A4)) = A4 ) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(S2,S4))
               => aa_set_complex_o(member_complex(aa_int_complex(J,A4)),minus_811609699411566653omplex(T2,T4)) )
           => ( ! [B4: complex] :
                  ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T2,T4))
                 => ( aa_int_complex(J,aa_complex_int(Ia,B4)) = B4 ) )
             => ( ! [B4: complex] :
                    ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(T2,T4))
                   => aa_set_int_o(member_int(aa_complex_int(Ia,B4)),minus_minus_set_int(S2,S4)) )
               => ( ! [A4: int] :
                      ( aa_set_int_o(member_int(A4),S4)
                     => ( aa_int_real(G,A4) = zero_zero_real ) )
                 => ( ! [B4: complex] :
                        ( aa_set_complex_o(member_complex(B4),T4)
                       => ( aa_complex_real(H,B4) = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( aa_set_int_o(member_int(A4),S2)
                         => ( aa_complex_real(H,aa_int_complex(J,A4)) = aa_int_real(G,A4) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8036_exp__gt__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xa)) ) ).

% exp_gt_one
tff(fact_8037_take__bit__signed__take__bit,axiom,
    ! [Ma: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( aa_int_int(bit_se2923211474154528505it_int(Ma),aa_int_int(bit_ri631733984087533419it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(Ma),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_8038_exp__minus__inverse,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xa))) = one_one_real ).

% exp_minus_inverse
tff(fact_8039_exp__minus__inverse,axiom,
    ! [Xa: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xa)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xa))) = one_one_complex ).

% exp_minus_inverse
tff(fact_8040_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(K),one_one_int)) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_8041_mask__Suc__double,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))) ).

% mask_Suc_double
tff(fact_8042_mask__Suc__double,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,Nb))) ).

% mask_Suc_double
tff(fact_8043_sum__nonneg__0,axiom,
    ! [Sb: set_real,F: fun_real_real,Ia: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,Sb) = zero_zero_real )
         => ( aa_set_real_o(member_real(Ia),Sb)
           => ( aa_real_real(F,Ia) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8044_sum__nonneg__0,axiom,
    ! [Sb: set_int,F: fun_int_real,Ia: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,Sb) = zero_zero_real )
         => ( aa_set_int_o(member_int(Ia),Sb)
           => ( aa_int_real(F,Ia) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8045_sum__nonneg__0,axiom,
    ! [Sb: set_complex,F: fun_complex_real,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,Sb) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(Ia),Sb)
           => ( aa_complex_real(F,Ia) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8046_sum__nonneg__0,axiom,
    ! [Sb: set_real,F: fun_real_rat,Ia: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_real_o(member_real(Ia),Sb)
           => ( aa_real_rat(F,Ia) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8047_sum__nonneg__0,axiom,
    ! [Sb: set_nat,F: fun_nat_rat,Ia: nat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(Ia),Sb)
           => ( aa_nat_rat(F,Ia) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8048_sum__nonneg__0,axiom,
    ! [Sb: set_int,F: fun_int_rat,Ia: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_int_o(member_int(Ia),Sb)
           => ( aa_int_rat(F,Ia) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8049_sum__nonneg__0,axiom,
    ! [Sb: set_complex,F: fun_complex_rat,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,Sb) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(Ia),Sb)
           => ( aa_complex_rat(F,Ia) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8050_sum__nonneg__0,axiom,
    ! [Sb: set_real,F: fun_real_nat,Ia: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,Sb) = zero_zero_nat )
         => ( aa_set_real_o(member_real(Ia),Sb)
           => ( aa_real_nat(F,Ia) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8051_sum__nonneg__0,axiom,
    ! [Sb: set_int,F: fun_int_nat,Ia: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,Sb) = zero_zero_nat )
         => ( aa_set_int_o(member_int(Ia),Sb)
           => ( aa_int_nat(F,Ia) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8052_sum__nonneg__0,axiom,
    ! [Sb: set_complex,F: fun_complex_nat,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,Sb) = zero_zero_nat )
         => ( aa_set_complex_o(member_complex(Ia),Sb)
           => ( aa_complex_nat(F,Ia) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8053_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_real,F: fun_real_real,B2: real,Ia: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,Sb) = B2 )
         => ( aa_set_real_o(member_real(Ia),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8054_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_int,F: fun_int_real,B2: real,Ia: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,Sb) = B2 )
         => ( aa_set_int_o(member_int(Ia),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8055_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_complex,F: fun_complex_real,B2: real,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,Sb) = B2 )
         => ( aa_set_complex_o(member_complex(Ia),Sb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8056_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_real,F: fun_real_rat,B2: rat,Ia: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,Sb) = B2 )
         => ( aa_set_real_o(member_real(Ia),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8057_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_nat,F: fun_nat_rat,B2: rat,Ia: nat] :
      ( aa_set_nat_o(finite_finite_nat,Sb)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,Sb) = B2 )
         => ( aa_set_nat_o(member_nat(Ia),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8058_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_int,F: fun_int_rat,B2: rat,Ia: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,Sb) = B2 )
         => ( aa_set_int_o(member_int(Ia),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8059_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_complex,F: fun_complex_rat,B2: rat,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,Sb) = B2 )
         => ( aa_set_complex_o(member_complex(Ia),Sb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8060_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_real,F: fun_real_nat,B2: nat,Ia: real] :
      ( aa_set_real_o(finite_finite_real,Sb)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,Sb) = B2 )
         => ( aa_set_real_o(member_real(Ia),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8061_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_int,F: fun_int_nat,B2: nat,Ia: int] :
      ( aa_set_int_o(finite_finite_int,Sb)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,Sb) = B2 )
         => ( aa_set_int_o(member_int(Ia),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8062_sum__nonneg__leq__bound,axiom,
    ! [Sb: set_complex,F: fun_complex_nat,B2: nat,Ia: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sb)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,Sb) = B2 )
         => ( aa_set_complex_o(member_complex(Ia),Sb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,Ia)),B2) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8063_sum_Ointer__restrict,axiom,
    ! [A2: set_real,G: fun_real_complex,B2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups5754745047067104278omplex(G,inf_inf_set_real(A2,B2)) = groups5754745047067104278omplex(aa_set2225337358267413879omplex(aTP_Lamm_jt(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8064_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_complex,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3049146728041665814omplex(G,inf_inf_set_int(A2,B2)) = groups3049146728041665814omplex(aa_set2721330520047917815omplex(aTP_Lamm_ju(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8065_sum_Ointer__restrict,axiom,
    ! [A2: set_real,G: fun_real_real,B2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,inf_inf_set_real(A2,B2)) = groups8097168146408367636l_real(aa_set8744029839670122869l_real(aTP_Lamm_jv(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8066_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_real,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,inf_inf_set_int(A2,B2)) = groups8778361861064173332t_real(aa_set6366552415685541621t_real(aTP_Lamm_jw(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8067_sum_Ointer__restrict,axiom,
    ! [A2: set_complex,G: fun_complex_real,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,inf_inf_set_complex(A2,B2)) = groups5808333547571424918x_real(aa_set904117813986905849x_real(aTP_Lamm_jx(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8068_sum_Ointer__restrict,axiom,
    ! [A2: set_real,G: fun_real_rat,B2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,inf_inf_set_real(A2,B2)) = groups1300246762558778688al_rat(aa_set4110029769364456865al_rat(aTP_Lamm_jy(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8069_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_rat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,inf_inf_set_int(A2,B2)) = groups3906332499630173760nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_jz(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8070_sum_Ointer__restrict,axiom,
    ! [A2: set_complex,G: fun_complex_rat,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,inf_inf_set_complex(A2,B2)) = groups5058264527183730370ex_rat(aa_set7482584248617139493ex_rat(aTP_Lamm_ka(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8071_sum_Ointer__restrict,axiom,
    ! [A2: set_real,G: fun_real_nat,B2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,inf_inf_set_real(A2,B2)) = groups1935376822645274424al_nat(aa_set7618258327757853593al_nat(aTP_Lamm_kb(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8072_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: fun_int_nat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,inf_inf_set_int(A2,B2)) = groups4541462559716669496nt_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_kc(G),B2),A2) ) ) ).

% sum.inter_restrict
tff(fact_8073_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups5754745047067104278omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kd(G)))) = groups5754745047067104278omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8074_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ke(G)))) = groups3049146728041665814omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8075_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kf(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8076_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kg(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8077_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kh(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8078_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ki(G)))) = groups1300246762558778688al_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8079_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kj(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8080_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kk(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8081_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_kl(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8082_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_km(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8083_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_8084_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(plus_plus_int(K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_8085_sum__pos2,axiom,
    ! [I5: set_real,Ia: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,Ia))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8086_sum__pos2,axiom,
    ! [I5: set_int,Ia: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,Ia))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8087_sum__pos2,axiom,
    ! [I5: set_complex,Ia: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(Ia),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,Ia))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8088_sum__pos2,axiom,
    ! [I5: set_real,Ia: real,F: fun_re6065469302365115155d_enat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,Ia))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_rea1194391308913012860d_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups2800946370649118462d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8089_sum__pos2,axiom,
    ! [I5: set_nat,Ia: nat,F: fun_na8797796813480665199d_enat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,Ia))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups7108830773950497114d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8090_sum__pos2,axiom,
    ! [I5: set_int,Ia: int,F: fun_in5914218760682845459d_enat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,Ia))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_int_Extended_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups4225252721152677374d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8091_sum__pos2,axiom,
    ! [I5: set_complex,Ia: complex,F: fun_co6496365337113880337d_enat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(Ia),I5)
       => ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,Ia))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le2932123472753598470d_enat,zero_z5237406670263579293d_enat),aa_com3683005661447327866d_enat(F,I2)) )
           => aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),groups1752964319039525884d_enat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8092_sum__pos2,axiom,
    ! [I5: set_real,Ia: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(Ia),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,Ia))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8093_sum__pos2,axiom,
    ! [I5: set_nat,Ia: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(Ia),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,Ia))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8094_sum__pos2,axiom,
    ! [I5: set_int,Ia: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(Ia),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,Ia))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8095_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8096_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8097_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(F,(I2))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8691415230153176458o_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8098_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8099_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8100_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8101_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(F,(I2))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups7872700643590313910_o_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8102_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8103_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8104_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_8105_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T2,S2))
             => ( aa_real_complex(G,X3) = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
           => ( groups5754745047067104278omplex(G,T2) = groups5754745047067104278omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8106_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_complex(G,X3) = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ( aa_int_complex(G,X3) = aa_int_complex(H,X3) ) )
           => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8107_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T2,S2))
             => ( aa_real_real(G,X3) = zero_zero_real ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8108_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_real(G,X3) = zero_zero_real ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ( aa_int_real(G,X3) = aa_int_real(H,X3) ) )
           => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8109_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8110_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
           => ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8111_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ( aa_int_rat(G,X3) = aa_int_rat(H,X3) ) )
           => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8112_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
           => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8113_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T2,S2))
             => ( aa_real_nat(G,X3) = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
           => ( groups1935376822645274424al_nat(G,T2) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8114_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_nat(G,X3) = zero_zero_nat ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ( aa_int_nat(G,X3) = aa_int_nat(H,X3) ) )
           => ( groups4541462559716669496nt_nat(G,T2) = groups4541462559716669496nt_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8115_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T2,S2))
             => ( aa_real_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
           => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8116_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_complex(H,X3) = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ( aa_int_complex(G,X3) = aa_int_complex(H,X3) ) )
           => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8117_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T2,S2))
             => ( aa_real_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8118_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ( aa_int_real(G,X3) = aa_int_real(H,X3) ) )
           => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8119_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(H,X3) = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8120_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8121_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ( aa_int_rat(G,X3) = aa_int_rat(H,X3) ) )
           => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8122_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(H,X3) = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),S2)
               => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
           => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8123_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X3: real] :
              ( aa_set_real_o(member_real(X3),minus_minus_set_real(T2,S2))
             => ( aa_real_nat(H,X3) = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
           => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8124_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_nat(H,X3) = zero_zero_nat ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ( aa_int_nat(G,X3) = aa_int_nat(H,X3) ) )
           => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8125_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_complex(G,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8126_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_real(G,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8127_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8128_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8129_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8130_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_nat(G,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T2) = groups4541462559716669496nt_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8131_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8132_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8133_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T2,S2))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,T2) = groups2073611262835488442omplex(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8134_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T2,S2))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T2) = groups2906978787729119204at_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8135_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_complex(G,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8136_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_real(G,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8137_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8138_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8139_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8140_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S2))
             => ( aa_int_nat(G,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8141_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8142_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8143_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T2,S2))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,S2) = groups2073611262835488442omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8144_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S2),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(T2,S2))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8145_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(C5,B2))
                 => ( aa_real_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C5) = groups5754745047067104278omplex(H,C5) )
               => ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8146_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C5,B2))
                 => ( aa_int_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G,C5) = groups3049146728041665814omplex(H,C5) )
               => ( groups3049146728041665814omplex(G,A2) = groups3049146728041665814omplex(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8147_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(C5,B2))
                 => ( aa_real_real(H,B4) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C5) = groups8097168146408367636l_real(H,C5) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8148_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C5,B2))
                 => ( aa_int_real(H,B4) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C5) = groups8778361861064173332t_real(H,C5) )
               => ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8149_sum_Osame__carrierI,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C5)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C5)
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C5,B2))
                 => ( aa_complex_real(H,B4) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C5) = groups5808333547571424918x_real(H,C5) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8150_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(C5,B2))
                 => ( aa_real_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C5) = groups1300246762558778688al_rat(H,C5) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8151_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C5,B2))
                 => ( aa_int_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C5) = groups3906332499630173760nt_rat(H,C5) )
               => ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8152_sum_Osame__carrierI,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C5)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C5)
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: complex] :
                  ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C5,B2))
                 => ( aa_complex_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C5) = groups5058264527183730370ex_rat(H,C5) )
               => ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8153_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(C5,B2))
                 => ( aa_real_nat(H,B4) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C5) = groups1935376822645274424al_nat(H,C5) )
               => ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8154_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C5,B2))
                 => ( aa_int_nat(H,B4) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,C5) = groups4541462559716669496nt_nat(H,C5) )
               => ( groups4541462559716669496nt_nat(G,A2) = groups4541462559716669496nt_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8155_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(C5,B2))
                 => ( aa_real_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) )
              <=> ( groups5754745047067104278omplex(G,C5) = groups5754745047067104278omplex(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8156_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_complex(G,A4) = zero_zero_complex ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C5,B2))
                 => ( aa_int_complex(H,B4) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G,A2) = groups3049146728041665814omplex(H,B2) )
              <=> ( groups3049146728041665814omplex(G,C5) = groups3049146728041665814omplex(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8157_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(C5,B2))
                 => ( aa_real_real(H,B4) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G,C5) = groups8097168146408367636l_real(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8158_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C5,B2))
                 => ( aa_int_real(H,B4) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) )
              <=> ( groups8778361861064173332t_real(G,C5) = groups8778361861064173332t_real(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8159_sum_Osame__carrier,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C5)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C5)
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_real(G,A4) = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C5,B2))
                 => ( aa_complex_real(H,B4) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G,C5) = groups5808333547571424918x_real(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8160_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(C5,B2))
                 => ( aa_real_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G,C5) = groups1300246762558778688al_rat(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8161_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C5,B2))
                 => ( aa_int_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) )
              <=> ( groups3906332499630173760nt_rat(G,C5) = groups3906332499630173760nt_rat(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8162_sum_Osame__carrier,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C5)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B2),C5)
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_rat(G,A4) = zero_zero_rat ) )
           => ( ! [B4: complex] :
                  ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(C5,B2))
                 => ( aa_complex_rat(H,B4) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) )
              <=> ( groups5058264527183730370ex_rat(G,C5) = groups5058264527183730370ex_rat(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8163_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B2),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( aa_set_real_o(member_real(B4),minus_minus_set_real(C5,B2))
                 => ( aa_real_nat(H,B4) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) )
              <=> ( groups1935376822645274424al_nat(G,C5) = groups1935376822645274424al_nat(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8164_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B2),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_nat(G,A4) = zero_zero_nat ) )
           => ( ! [B4: int] :
                  ( aa_set_int_o(member_int(B4),minus_minus_set_int(C5,B2))
                 => ( aa_int_nat(H,B4) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,A2) = groups4541462559716669496nt_nat(H,B2) )
              <=> ( groups4541462559716669496nt_nat(G,C5) = groups4541462559716669496nt_nat(H,C5) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8165_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8166_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8167_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8168_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2))),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8169_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8170_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8171_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B2),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(plus_plus_int(groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8172_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups2906978787729119204at_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2))),groups2906978787729119204at_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8173_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups3539618377306564664at_int(G,A2) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8174_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B2),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups4538972089207619220nt_int(G,A2) = aa_int_int(plus_plus_int(groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_8175_even__or__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1080825931792720795nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_8176_even__or__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_8177_even__or__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_or_iff
tff(fact_8178_sum_Omono__neutral__cong,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(finite_finite_real,S2)
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),minus_minus_set_real(T2,S2))
             => ( aa_real_complex(H,I2) = zero_zero_complex ) )
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),minus_minus_set_real(S2,T2))
               => ( aa_real_complex(G,I2) = zero_zero_complex ) )
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),inf_inf_set_real(S2,T2))
                 => ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
             => ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8179_sum_Omono__neutral__cong,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(finite_finite_int,S2)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T2,S2))
             => ( aa_int_complex(H,I2) = zero_zero_complex ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S2,T2))
               => ( aa_int_complex(G,I2) = zero_zero_complex ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),inf_inf_set_int(S2,T2))
                 => ( aa_int_complex(G,X3) = aa_int_complex(H,X3) ) )
             => ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8180_sum_Omono__neutral__cong,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(finite_finite_real,S2)
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),minus_minus_set_real(T2,S2))
             => ( aa_real_real(H,I2) = zero_zero_real ) )
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),minus_minus_set_real(S2,T2))
               => ( aa_real_real(G,I2) = zero_zero_real ) )
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),inf_inf_set_real(S2,T2))
                 => ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
             => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8181_sum_Omono__neutral__cong,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(finite_finite_int,S2)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T2,S2))
             => ( aa_int_real(H,I2) = zero_zero_real ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S2,T2))
               => ( aa_int_real(G,I2) = zero_zero_real ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),inf_inf_set_int(S2,T2))
                 => ( aa_int_real(G,X3) = aa_int_real(H,X3) ) )
             => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8182_sum_Omono__neutral__cong,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(H,I2) = zero_zero_real ) )
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),minus_811609699411566653omplex(S2,T2))
               => ( aa_complex_real(G,I2) = zero_zero_real ) )
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),inf_inf_set_complex(S2,T2))
                 => ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
             => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8183_sum_Omono__neutral__cong,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(finite_finite_real,S2)
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(H,I2) = zero_zero_rat ) )
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),minus_minus_set_real(S2,T2))
               => ( aa_real_rat(G,I2) = zero_zero_rat ) )
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),inf_inf_set_real(S2,T2))
                 => ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
             => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8184_sum_Omono__neutral__cong,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(finite_finite_int,S2)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T2,S2))
             => ( aa_int_rat(H,I2) = zero_zero_rat ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S2,T2))
               => ( aa_int_rat(G,I2) = zero_zero_rat ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),inf_inf_set_int(S2,T2))
                 => ( aa_int_rat(G,X3) = aa_int_rat(H,X3) ) )
             => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8185_sum_Omono__neutral__cong,axiom,
    ! [T2: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(H,I2) = zero_zero_rat ) )
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),minus_811609699411566653omplex(S2,T2))
               => ( aa_complex_rat(G,I2) = zero_zero_rat ) )
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),inf_inf_set_complex(S2,T2))
                 => ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
             => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8186_sum_Omono__neutral__cong,axiom,
    ! [T2: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(finite_finite_real,S2)
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),minus_minus_set_real(T2,S2))
             => ( aa_real_nat(H,I2) = zero_zero_nat ) )
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),minus_minus_set_real(S2,T2))
               => ( aa_real_nat(G,I2) = zero_zero_nat ) )
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),inf_inf_set_real(S2,T2))
                 => ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
             => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8187_sum_Omono__neutral__cong,axiom,
    ! [T2: set_int,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(finite_finite_int,S2)
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),minus_minus_set_int(T2,S2))
             => ( aa_int_nat(H,I2) = zero_zero_nat ) )
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),minus_minus_set_int(S2,T2))
               => ( aa_int_nat(G,I2) = zero_zero_nat ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),inf_inf_set_int(S2,T2))
                 => ( aa_int_nat(G,X3) = aa_int_nat(H,X3) ) )
             => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(H,T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_8188_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,sup_sup_set_int(A2,B2))),groups8778361861064173332t_real(G,inf_inf_set_int(A2,B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,A2)),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8189_sum_Ounion__inter,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,sup_sup_set_complex(A2,B2))),groups5808333547571424918x_real(G,inf_inf_set_complex(A2,B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,A2)),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8190_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,sup_sup_set_int(A2,B2))),groups3906332499630173760nt_rat(G,inf_inf_set_int(A2,B2))) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,A2)),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8191_sum_Ounion__inter,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,sup_sup_set_complex(A2,B2))),groups5058264527183730370ex_rat(G,inf_inf_set_complex(A2,B2))) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,A2)),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8192_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,sup_sup_set_int(A2,B2))),groups4541462559716669496nt_nat(G,inf_inf_set_int(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,A2)),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8193_sum_Ounion__inter,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,sup_sup_set_complex(A2,B2))),groups5693394587270226106ex_nat(G,inf_inf_set_complex(A2,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,A2)),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8194_sum_Ounion__inter,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( aa_int_int(plus_plus_int(groups5690904116761175830ex_int(G,sup_sup_set_complex(A2,B2))),groups5690904116761175830ex_int(G,inf_inf_set_complex(A2,B2))) = aa_int_int(plus_plus_int(groups5690904116761175830ex_int(G,A2)),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8195_sum_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,sup_sup_set_nat(A2,B2))),groups2906978787729119204at_rat(G,inf_inf_set_nat(A2,B2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,A2)),groups2906978787729119204at_rat(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8196_sum_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,sup_sup_set_nat(A2,B2))),groups3539618377306564664at_int(G,inf_inf_set_nat(A2,B2))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,A2)),groups3539618377306564664at_int(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8197_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( aa_int_int(plus_plus_int(groups4538972089207619220nt_int(G,sup_sup_set_int(A2,B2))),groups4538972089207619220nt_int(G,inf_inf_set_int(A2,B2))) = aa_int_int(plus_plus_int(groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(G,B2)) ) ) ) ).

% sum.union_inter
tff(fact_8198_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_real,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,inf_inf_set_int(A2,B2))),groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8199_sum_OInt__Diff,axiom,
    ! [A2: set_complex,G: fun_complex_real,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,inf_inf_set_complex(A2,B2))),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8200_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_rat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,inf_inf_set_int(A2,B2))),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8201_sum_OInt__Diff,axiom,
    ! [A2: set_complex,G: fun_complex_rat,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,inf_inf_set_complex(A2,B2))),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8202_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_nat,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,inf_inf_set_int(A2,B2))),groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8203_sum_OInt__Diff,axiom,
    ! [A2: set_complex,G: fun_complex_nat,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,inf_inf_set_complex(A2,B2))),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8204_sum_OInt__Diff,axiom,
    ! [A2: set_complex,G: fun_complex_int,B2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(plus_plus_int(groups5690904116761175830ex_int(G,inf_inf_set_complex(A2,B2))),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8205_sum_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_rat,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,inf_inf_set_nat(A2,B2))),groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8206_sum_OInt__Diff,axiom,
    ! [A2: set_nat,G: fun_nat_int,B2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups3539618377306564664at_int(G,A2) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,inf_inf_set_nat(A2,B2))),groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8207_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: fun_int_int,B2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4538972089207619220nt_int(G,A2) = aa_int_int(plus_plus_int(groups4538972089207619220nt_int(G,inf_inf_set_int(A2,B2))),groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))) ) ) ).

% sum.Int_Diff
tff(fact_8208_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,Xa: code_integer,Y: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,Xa) = zero_z3403309356797280102nteger )
     => ( ( bit_se1080825931792720795nteger(A,Xa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
       => ( ( bit_se3949692690581998587nteger(A,Y) = zero_z3403309356797280102nteger )
         => ( ( bit_se1080825931792720795nteger(A,Y) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( Xa = Y ) ) ) ) ) ).

% bit.complement_unique
tff(fact_8209_bit_Ocomplement__unique,axiom,
    ! [A: int,Xa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),Xa) = zero_zero_int )
     => ( ( aa_int_int(bit_se1409905431419307370or_int(A),Xa) = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( aa_int_int(bit_se725231765392027082nd_int(A),Y) = zero_zero_int )
         => ( ( aa_int_int(bit_se1409905431419307370or_int(A),Y) = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( Xa = Y ) ) ) ) ) ).

% bit.complement_unique
tff(fact_8210_exp__ge__add__one__self__aux,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(exp_real,Xa)) ) ).

% exp_ge_add_one_self_aux
tff(fact_8211_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real))
          & ( aa_real_real(exp_real,X3) = Y ) ) ) ).

% lemma_exp_total
tff(fact_8212_ln__ge__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xa) ) ) ).

% ln_ge_iff
tff(fact_8213_sum_OIf__cases,axiom,
    ! [A2: set_real,P: fun_real_o,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_kn(P),H),G),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8097168146408367636l_real(H,inf_inf_set_real(A2,collect_real(P)))),groups8097168146408367636l_real(G,inf_inf_set_real(A2,uminus612125837232591019t_real(collect_real(P))))) ) ) ).

% sum.If_cases
tff(fact_8214_sum_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ko(P),H),G),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(H,inf_inf_set_int(A2,collect_int(P)))),groups8778361861064173332t_real(G,inf_inf_set_int(A2,uminus1532241313380277803et_int(collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8215_sum_OIf__cases,axiom,
    ! [A2: set_complex,P: fun_complex_o,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kp(P),H),G),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(H,inf_inf_set_complex(A2,collect_complex(P)))),groups5808333547571424918x_real(G,inf_inf_set_complex(A2,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8216_sum_OIf__cases,axiom,
    ! [A2: set_real,P: fun_real_o,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_kq(P),H),G),A2) = aa_rat_rat(plus_plus_rat(groups1300246762558778688al_rat(H,inf_inf_set_real(A2,collect_real(P)))),groups1300246762558778688al_rat(G,inf_inf_set_real(A2,uminus612125837232591019t_real(collect_real(P))))) ) ) ).

% sum.If_cases
tff(fact_8217_sum_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_kr(P),H),G),A2) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(H,inf_inf_set_int(A2,collect_int(P)))),groups3906332499630173760nt_rat(G,inf_inf_set_int(A2,uminus1532241313380277803et_int(collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8218_sum_OIf__cases,axiom,
    ! [A2: set_complex,P: fun_complex_o,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_ks(P),H),G),A2) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(H,inf_inf_set_complex(A2,collect_complex(P)))),groups5058264527183730370ex_rat(G,inf_inf_set_complex(A2,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8219_sum_OIf__cases,axiom,
    ! [A2: set_real,P: fun_real_o,H: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_kt(P),H),G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups1935376822645274424al_nat(H,inf_inf_set_real(A2,collect_real(P)))),groups1935376822645274424al_nat(G,inf_inf_set_real(A2,uminus612125837232591019t_real(collect_real(P))))) ) ) ).

% sum.If_cases
tff(fact_8220_sum_OIf__cases,axiom,
    ! [A2: set_int,P: fun_int_o,H: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ku(P),H),G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(H,inf_inf_set_int(A2,collect_int(P)))),groups4541462559716669496nt_nat(G,inf_inf_set_int(A2,uminus1532241313380277803et_int(collect_int(P))))) ) ) ).

% sum.If_cases
tff(fact_8221_sum_OIf__cases,axiom,
    ! [A2: set_complex,P: fun_complex_o,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_kv(P),H),G),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(H,inf_inf_set_complex(A2,collect_complex(P)))),groups5693394587270226106ex_nat(G,inf_inf_set_complex(A2,uminus8566677241136511917omplex(collect_complex(P))))) ) ) ).

% sum.If_cases
tff(fact_8222_sum_OIf__cases,axiom,
    ! [A2: set_real,P: fun_real_o,H: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kw(P),H),G),A2) = aa_int_int(plus_plus_int(groups1932886352136224148al_int(H,inf_inf_set_real(A2,collect_real(P)))),groups1932886352136224148al_int(G,inf_inf_set_real(A2,uminus612125837232591019t_real(collect_real(P))))) ) ) ).

% sum.If_cases
tff(fact_8223_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_8224_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_8225_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(Nb,A) = modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_8226_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_8227_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_8228_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [B4: real] :
              ( aa_set_real_o(member_real(B4),minus_minus_set_real(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8229_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( ! [B4: int] :
              ( aa_set_int_o(member_int(B4),minus_minus_set_int(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,B4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8230_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [B4: complex] :
              ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B2,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8231_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [B4: real] :
              ( aa_set_real_o(member_real(B4),minus_minus_set_real(B2,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8232_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( ! [B4: int] :
              ( aa_set_int_o(member_int(B4),minus_minus_set_int(B2,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8233_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [B4: complex] :
              ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B2,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8234_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [B4: real] :
              ( aa_set_real_o(member_real(B4),minus_minus_set_real(B2,A2))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8235_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( ! [B4: int] :
              ( aa_set_int_o(member_int(B4),minus_minus_set_int(B2,A2))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,B4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8236_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( ! [B4: complex] :
              ( aa_set_complex_o(member_complex(B4),minus_811609699411566653omplex(B2,A2))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8237_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( ! [B4: real] :
              ( aa_set_real_o(member_real(B4),minus_minus_set_real(B2,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,B4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B2)) ) ) ) ).

% sum_mono2
tff(fact_8238_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma) = Ma )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_8239_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_8240_take__bit__nat__eq__self,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma) = Ma ) ) ).

% take_bit_nat_eq_self
tff(fact_8241_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),inf_inf_set_int(A2,B2))
             => ( aa_int_complex(G,X3) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,sup_sup_set_int(A2,B2)) = aa_complex_complex(plus_plus_complex(groups3049146728041665814omplex(G,A2)),groups3049146728041665814omplex(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8242_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),inf_inf_set_int(A2,B2))
             => ( aa_int_real(G,X3) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,sup_sup_set_int(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,A2)),groups8778361861064173332t_real(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8243_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),inf_inf_set_complex(A2,B2))
             => ( aa_complex_real(G,X3) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,sup_sup_set_complex(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,A2)),groups5808333547571424918x_real(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8244_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),inf_inf_set_int(A2,B2))
             => ( aa_int_rat(G,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,sup_sup_set_int(A2,B2)) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,A2)),groups3906332499630173760nt_rat(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8245_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),inf_inf_set_complex(A2,B2))
             => ( aa_complex_rat(G,X3) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,sup_sup_set_complex(A2,B2)) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,A2)),groups5058264527183730370ex_rat(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8246_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),inf_inf_set_int(A2,B2))
             => ( aa_int_nat(G,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,sup_sup_set_int(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,A2)),groups4541462559716669496nt_nat(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8247_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),inf_inf_set_complex(A2,B2))
             => ( aa_complex_nat(G,X3) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,sup_sup_set_complex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,A2)),groups5693394587270226106ex_nat(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8248_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ! [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),inf_inf_set_complex(A2,B2))
             => ( aa_complex_int(G,X3) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,sup_sup_set_complex(A2,B2)) = aa_int_int(plus_plus_int(groups5690904116761175830ex_int(G,A2)),groups5690904116761175830ex_int(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8249_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),inf_inf_set_nat(A2,B2))
             => ( aa_nat_complex(G,X3) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,sup_sup_set_nat(A2,B2)) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,A2)),groups2073611262835488442omplex(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8250_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),inf_inf_set_nat(A2,B2))
             => ( aa_nat_rat(G,X3) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,sup_sup_set_nat(A2,B2)) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,A2)),groups2906978787729119204at_rat(G,B2)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8251_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(G,insert_complex(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xa)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8252_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_rat,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(G,insert_complex(Xa,A2)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xa)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8253_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_nat,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5693394587270226106ex_nat(G,insert_complex(Xa,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xa)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8254_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_int,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(G,insert_complex(Xa,A2)) = aa_int_int(plus_plus_int(aa_complex_int(G,Xa)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_8255_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(G,insert_real(Xa,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xa)),groups8097168146408367636l_real(G,minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_8256_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_rat,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(G,insert_real(Xa,A2)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xa)),groups1300246762558778688al_rat(G,minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_8257_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_nat,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1935376822645274424al_nat(G,insert_real(Xa,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,Xa)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_8258_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_int,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1932886352136224148al_int(G,insert_real(Xa,A2)) = aa_int_int(plus_plus_int(aa_real_int(G,Xa)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ).

% sum.insert_remove
tff(fact_8259_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_real,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8691415230153176458o_real(G,insert_o((Xa),A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_o_real(G,(Xa))),groups8691415230153176458o_real(G,minus_minus_set_o(A2,insert_o((Xa),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8260_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: fun_o_rat,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups7872700643590313910_o_rat(G,insert_o((Xa),A2)) = aa_rat_rat(plus_plus_rat(aa_o_rat(G,(Xa))),groups7872700643590313910_o_rat(G,minus_minus_set_o(A2,insert_o((Xa),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8261_sum_Oremove,axiom,
    ! [A2: set_complex,Xa: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xa)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8262_sum_Oremove,axiom,
    ! [A2: set_complex,Xa: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xa)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8263_sum_Oremove,axiom,
    ! [A2: set_complex,Xa: complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xa)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8264_sum_Oremove,axiom,
    ! [A2: set_complex,Xa: complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xa),A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(plus_plus_int(aa_complex_int(G,Xa)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(Xa,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_8265_sum_Oremove,axiom,
    ! [A2: set_real,Xa: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( groups8097168146408367636l_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xa)),groups8097168146408367636l_real(G,minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8266_sum_Oremove,axiom,
    ! [A2: set_real,Xa: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( groups1300246762558778688al_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xa)),groups1300246762558778688al_rat(G,minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8267_sum_Oremove,axiom,
    ! [A2: set_real,Xa: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( groups1935376822645274424al_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,Xa)),groups1935376822645274424al_nat(G,minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8268_sum_Oremove,axiom,
    ! [A2: set_real,Xa: real,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xa),A2)
       => ( groups1932886352136224148al_int(G,A2) = aa_int_int(plus_plus_int(aa_real_int(G,Xa)),groups1932886352136224148al_int(G,minus_minus_set_real(A2,insert_real(Xa,bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_8269_sum_Oremove,axiom,
    ! [A2: set_o,Xa: $o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( groups8691415230153176458o_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_o_real(G,(Xa))),groups8691415230153176458o_real(G,minus_minus_set_o(A2,insert_o((Xa),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8270_sum_Oremove,axiom,
    ! [A2: set_o,Xa: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(member_o((Xa)),A2)
       => ( groups7872700643590313910_o_rat(G,A2) = aa_rat_rat(plus_plus_rat(aa_o_rat(G,(Xa))),groups7872700643590313910_o_rat(G,minus_minus_set_o(A2,insert_o((Xa),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8271_sum__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_real,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),aa_complex_real(F,A)),groups5808333547571424918x_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8272_sum__diff1,axiom,
    ! [A2: set_real,F: fun_real_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups8097168146408367636l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(F,A2)),aa_real_real(F,A)),groups8097168146408367636l_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8273_sum__diff1,axiom,
    ! [A2: set_o,F: fun_o_real,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups8691415230153176458o_real(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8691415230153176458o_real(F,A2)),aa_o_real(F,(A))),groups8691415230153176458o_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8274_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_real,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),aa_int_real(F,A)),groups8778361861064173332t_real(F,A2)) ) ) ).

% sum_diff1
tff(fact_8275_sum__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A2)),aa_complex_rat(F,A)),groups5058264527183730370ex_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8276_sum__diff1,axiom,
    ! [A2: set_real,F: fun_real_rat,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(F,A2)),aa_real_rat(F,A)),groups1300246762558778688al_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8277_sum__diff1,axiom,
    ! [A2: set_o,F: fun_o_rat,A: $o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( groups7872700643590313910_o_rat(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_rat_rat(minus_minus_rat(groups7872700643590313910_o_rat(F,A2)),aa_o_rat(F,(A))),groups7872700643590313910_o_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8278_sum__diff1,axiom,
    ! [A2: set_int,F: fun_int_rat,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A2)),aa_int_rat(F,A)),groups3906332499630173760nt_rat(F,A2)) ) ) ).

% sum_diff1
tff(fact_8279_sum__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_int,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_int_int(minus_minus_int(groups5690904116761175830ex_int(F,A2)),aa_complex_int(F,A)),groups5690904116761175830ex_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_8280_sum__diff1,axiom,
    ! [A2: set_real,F: fun_real_int,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1932886352136224148al_int(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_int_int(minus_minus_int(groups1932886352136224148al_int(F,A2)),aa_real_int(F,A)),groups1932886352136224148al_int(F,A2)) ) ) ).

% sum_diff1
tff(fact_8281_sum__Un,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups8778361861064173332t_real(F,sup_sup_set_int(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2))),groups8778361861064173332t_real(F,inf_inf_set_int(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8282_sum__Un,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5808333547571424918x_real(F,sup_sup_set_complex(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2))),groups5808333547571424918x_real(F,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8283_sum__Un,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups3906332499630173760nt_rat(F,sup_sup_set_int(A2,B2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))),groups3906332499630173760nt_rat(F,inf_inf_set_int(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8284_sum__Un,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5058264527183730370ex_rat(F,sup_sup_set_complex(A2,B2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2))),groups5058264527183730370ex_rat(F,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8285_sum__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups2906978787729119204at_rat(F,sup_sup_set_nat(A2,B2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B2))),groups2906978787729119204at_rat(F,inf_inf_set_nat(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8286_sum__Un,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5690904116761175830ex_int(F,sup_sup_set_complex(A2,B2)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2))),groups5690904116761175830ex_int(F,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8287_sum__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups3539618377306564664at_int(F,sup_sup_set_nat(A2,B2)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2))),groups3539618377306564664at_int(F,inf_inf_set_nat(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8288_sum__Un,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4538972089207619220nt_int(F,sup_sup_set_int(A2,B2)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B2))),groups4538972089207619220nt_int(F,inf_inf_set_int(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8289_sum__Un,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups7754918857620584856omplex(F,sup_sup_set_complex(A2,B2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(F,B2))),groups7754918857620584856omplex(F,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8290_sum__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups6591440286371151544t_real(F,sup_sup_set_nat(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(F,B2))),groups6591440286371151544t_real(F,inf_inf_set_nat(A2,B2))) ) ) ) ).

% sum_Un
tff(fact_8291_sum_Ounion__disjoint,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ( inf_inf_set_complex(A2,B2) = bot_bot_set_complex )
         => ( groups5808333547571424918x_real(G,sup_sup_set_complex(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,A2)),groups5808333547571424918x_real(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8292_sum_Ounion__disjoint,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ( inf_inf_set_complex(A2,B2) = bot_bot_set_complex )
         => ( groups5058264527183730370ex_rat(G,sup_sup_set_complex(A2,B2)) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,A2)),groups5058264527183730370ex_rat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8293_sum_Ounion__disjoint,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ( inf_inf_set_complex(A2,B2) = bot_bot_set_complex )
         => ( groups5693394587270226106ex_nat(G,sup_sup_set_complex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,A2)),groups5693394587270226106ex_nat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8294_sum_Ounion__disjoint,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( ( inf_inf_set_complex(A2,B2) = bot_bot_set_complex )
         => ( groups5690904116761175830ex_int(G,sup_sup_set_complex(A2,B2)) = aa_int_int(plus_plus_int(groups5690904116761175830ex_int(G,A2)),groups5690904116761175830ex_int(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8295_sum_Ounion__disjoint,axiom,
    ! [A2: set_real,B2: set_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(finite_finite_real,B2)
       => ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
         => ( groups8097168146408367636l_real(G,sup_sup_set_real(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8097168146408367636l_real(G,A2)),groups8097168146408367636l_real(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8296_sum_Ounion__disjoint,axiom,
    ! [A2: set_real,B2: set_real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(finite_finite_real,B2)
       => ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
         => ( groups1300246762558778688al_rat(G,sup_sup_set_real(A2,B2)) = aa_rat_rat(plus_plus_rat(groups1300246762558778688al_rat(G,A2)),groups1300246762558778688al_rat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8297_sum_Ounion__disjoint,axiom,
    ! [A2: set_real,B2: set_real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(finite_finite_real,B2)
       => ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
         => ( groups1935376822645274424al_nat(G,sup_sup_set_real(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups1935376822645274424al_nat(G,A2)),groups1935376822645274424al_nat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8298_sum_Ounion__disjoint,axiom,
    ! [A2: set_real,B2: set_real,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(finite_finite_real,B2)
       => ( ( inf_inf_set_real(A2,B2) = bot_bot_set_real )
         => ( groups1932886352136224148al_int(G,sup_sup_set_real(A2,B2)) = aa_int_int(plus_plus_int(groups1932886352136224148al_int(G,A2)),groups1932886352136224148al_int(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8299_sum_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
         => ( groups8691415230153176458o_real(G,sup_sup_set_o(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8691415230153176458o_real(G,A2)),groups8691415230153176458o_real(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8300_sum_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( aa_set_o_o(finite_finite_o,B2)
       => ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
         => ( groups7872700643590313910_o_rat(G,sup_sup_set_o(A2,B2)) = aa_rat_rat(plus_plus_rat(groups7872700643590313910_o_rat(G,A2)),groups7872700643590313910_o_rat(G,B2)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_8301_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups8778361861064173332t_real(G,sup_sup_set_int(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,minus_minus_set_int(B2,A2)))),groups8778361861064173332t_real(G,inf_inf_set_int(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8302_sum_Ounion__diff2,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5808333547571424918x_real(G,sup_sup_set_complex(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,minus_811609699411566653omplex(B2,A2)))),groups5808333547571424918x_real(G,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8303_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups3906332499630173760nt_rat(G,sup_sup_set_int(A2,B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(G,minus_minus_set_int(B2,A2)))),groups3906332499630173760nt_rat(G,inf_inf_set_int(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8304_sum_Ounion__diff2,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5058264527183730370ex_rat(G,sup_sup_set_complex(A2,B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2))),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(B2,A2)))),groups5058264527183730370ex_rat(G,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8305_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4541462559716669496nt_nat(G,sup_sup_set_int(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,minus_minus_set_int(B2,A2)))),groups4541462559716669496nt_nat(G,inf_inf_set_int(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8306_sum_Ounion__diff2,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5693394587270226106ex_nat(G,sup_sup_set_complex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(B2,A2)))),groups5693394587270226106ex_nat(G,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8307_sum_Ounion__diff2,axiom,
    ! [A2: set_complex,B2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5690904116761175830ex_int(G,sup_sup_set_complex(A2,B2)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(B2,A2)))),groups5690904116761175830ex_int(G,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8308_sum_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups2906978787729119204at_rat(G,sup_sup_set_nat(A2,B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2))),groups2906978787729119204at_rat(G,minus_minus_set_nat(B2,A2)))),groups2906978787729119204at_rat(G,inf_inf_set_nat(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8309_sum_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( groups3539618377306564664at_int(G,sup_sup_set_nat(A2,B2)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(G,minus_minus_set_nat(B2,A2)))),groups3539618377306564664at_int(G,inf_inf_set_nat(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8310_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4538972089207619220nt_int(G,sup_sup_set_int(A2,B2)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(G,minus_minus_set_int(B2,A2)))),groups4538972089207619220nt_int(G,inf_inf_set_int(A2,B2))) ) ) ) ).

% sum.union_diff2
tff(fact_8311_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,sup_sup_set_int(A2,B2))
     => ( groups8778361861064173332t_real(F,sup_sup_set_int(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(F,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(F,minus_minus_set_int(B2,A2)))),groups8778361861064173332t_real(F,inf_inf_set_int(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8312_sum__Un2,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A2,B2))
     => ( groups5808333547571424918x_real(F,sup_sup_set_complex(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(F,minus_811609699411566653omplex(B2,A2)))),groups5808333547571424918x_real(F,inf_inf_set_complex(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8313_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,sup_sup_set_int(A2,B2))
     => ( groups3906332499630173760nt_rat(F,sup_sup_set_int(A2,B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(F,minus_minus_set_int(B2,A2)))),groups3906332499630173760nt_rat(F,inf_inf_set_int(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8314_sum__Un2,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A2,B2))
     => ( groups5058264527183730370ex_rat(F,sup_sup_set_complex(A2,B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,B2))),groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(B2,A2)))),groups5058264527183730370ex_rat(F,inf_inf_set_complex(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8315_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,sup_sup_set_int(A2,B2))
     => ( groups4541462559716669496nt_nat(F,sup_sup_set_int(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(F,minus_minus_set_int(B2,A2)))),groups4541462559716669496nt_nat(F,inf_inf_set_int(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8316_sum__Un2,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A2,B2))
     => ( groups5693394587270226106ex_nat(F,sup_sup_set_complex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(B2,A2)))),groups5693394587270226106ex_nat(F,inf_inf_set_complex(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8317_sum__Un2,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,sup_sup_set_complex(A2,B2))
     => ( groups5690904116761175830ex_int(F,sup_sup_set_complex(A2,B2)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(F,minus_811609699411566653omplex(B2,A2)))),groups5690904116761175830ex_int(F,inf_inf_set_complex(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8318_sum__Un2,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,sup_sup_set_nat(A2,B2))
     => ( groups2906978787729119204at_rat(F,sup_sup_set_nat(A2,B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,B2))),groups2906978787729119204at_rat(F,minus_minus_set_nat(B2,A2)))),groups2906978787729119204at_rat(F,inf_inf_set_nat(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8319_sum__Un2,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,sup_sup_set_nat(A2,B2))
     => ( groups3539618377306564664at_int(F,sup_sup_set_nat(A2,B2)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(groups3539618377306564664at_int(F,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(F,minus_minus_set_nat(B2,A2)))),groups3539618377306564664at_int(F,inf_inf_set_nat(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8320_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,sup_sup_set_int(A2,B2))
     => ( groups4538972089207619220nt_int(F,sup_sup_set_int(A2,B2)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(groups4538972089207619220nt_int(F,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(F,minus_minus_set_int(B2,A2)))),groups4538972089207619220nt_int(F,inf_inf_set_int(A2,B2))) ) ) ).

% sum_Un2
tff(fact_8321_take__bit__nat__def,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_8322_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% exp_le
tff(fact_8323_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_8324_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(modulo_modulo_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_def
tff(fact_8325_sum__div__partition,axiom,
    ! [A2: set_real,F: fun_re6978948838062957830nteger,B: code_integer] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( divide6298287555418463151nteger(groups7713935264441627589nteger(F,A2),B) = plus_p5714425477246183910nteger(groups7713935264441627589nteger(aa_Cod7053582236991556129nteger(aTP_Lamm_kx(F),B),inf_inf_set_real(A2,collect_real(aa_Cod1315720040876898922real_o(aTP_Lamm_ky(F),B)))),divide6298287555418463151nteger(groups7713935264441627589nteger(F,inf_inf_set_real(A2,collect_real(aa_Cod1315720040876898922real_o(aTP_Lamm_kz(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8326_sum__div__partition,axiom,
    ! [A2: set_int,F: fun_int_Code_integer,B: code_integer] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( divide6298287555418463151nteger(groups7873554091576472773nteger(F,A2),B) = plus_p5714425477246183910nteger(groups7873554091576472773nteger(aa_Cod1949805079358391713nteger(aTP_Lamm_la(F),B),inf_inf_set_int(A2,collect_int(aa_Cod8477617969808203498_int_o(aTP_Lamm_lb(F),B)))),divide6298287555418463151nteger(groups7873554091576472773nteger(F,inf_inf_set_int(A2,collect_int(aa_Cod8477617969808203498_int_o(aTP_Lamm_lc(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8327_sum__div__partition,axiom,
    ! [A2: set_complex,F: fun_co8153928105989009288nteger,B: code_integer] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( divide6298287555418463151nteger(groups6621422865394947399nteger(F,A2),B) = plus_p5714425477246183910nteger(groups6621422865394947399nteger(aa_Cod1660996421839551523nteger(aTP_Lamm_ld(F),B),inf_inf_set_complex(A2,collect_complex(aa_Cod7908732550325689320plex_o(aTP_Lamm_le(F),B)))),divide6298287555418463151nteger(groups6621422865394947399nteger(F,inf_inf_set_complex(A2,collect_complex(aa_Cod7908732550325689320plex_o(aTP_Lamm_lf(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8328_sum__div__partition,axiom,
    ! [A2: set_nat,F: fun_nat_Code_integer,B: code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( divide6298287555418463151nteger(groups7501900531339628137nteger(F,A2),B) = plus_p5714425477246183910nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_lg(F),B),inf_inf_set_nat(A2,collect_nat(aa_Cod1075288278386115654_nat_o(aTP_Lamm_lh(F),B)))),divide6298287555418463151nteger(groups7501900531339628137nteger(F,inf_inf_set_nat(A2,collect_nat(aa_Cod1075288278386115654_nat_o(aTP_Lamm_li(F),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_8329_sum__div__partition,axiom,
    ! [A2: set_real,F: fun_real_nat,B: nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups1935376822645274424al_nat(F,A2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_lj(F),B),inf_inf_set_real(A2,collect_real(aa_nat_fun_real_o(aTP_Lamm_lk(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups1935376822645274424al_nat(F,inf_inf_set_real(A2,collect_real(aa_nat_fun_real_o(aTP_Lamm_ll(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8330_sum__div__partition,axiom,
    ! [A2: set_int,F: fun_int_nat,B: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4541462559716669496nt_nat(F,A2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_lm(F),B),inf_inf_set_int(A2,collect_int(aa_nat_fun_int_o(aTP_Lamm_ln(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4541462559716669496nt_nat(F,inf_inf_set_int(A2,collect_int(aa_nat_fun_int_o(aTP_Lamm_lo(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8331_sum__div__partition,axiom,
    ! [A2: set_complex,F: fun_complex_nat,B: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups5693394587270226106ex_nat(F,A2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_lp(F),B),inf_inf_set_complex(A2,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_lq(F),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups5693394587270226106ex_nat(F,inf_inf_set_complex(A2,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_lr(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8332_sum__div__partition,axiom,
    ! [A2: set_real,F: fun_real_int,B: int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_int_int(divide_divide_int(groups1932886352136224148al_int(F,A2)),B) = aa_int_int(plus_plus_int(groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_ls(F),B),inf_inf_set_real(A2,collect_real(aa_int_fun_real_o(aTP_Lamm_lt(F),B))))),aa_int_int(divide_divide_int(groups1932886352136224148al_int(F,inf_inf_set_real(A2,collect_real(aa_int_fun_real_o(aTP_Lamm_lu(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8333_sum__div__partition,axiom,
    ! [A2: set_complex,F: fun_complex_int,B: int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_int_int(divide_divide_int(groups5690904116761175830ex_int(F,A2)),B) = aa_int_int(plus_plus_int(groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_lv(F),B),inf_inf_set_complex(A2,collect_complex(aa_int_fun_complex_o(aTP_Lamm_lw(F),B))))),aa_int_int(divide_divide_int(groups5690904116761175830ex_int(F,inf_inf_set_complex(A2,collect_complex(aa_int_fun_complex_o(aTP_Lamm_lx(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8334_sum__div__partition,axiom,
    ! [A2: set_nat,F: fun_nat_int,B: int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_int_int(divide_divide_int(groups3539618377306564664at_int(F,A2)),B) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ly(F),B),inf_inf_set_nat(A2,collect_nat(aa_int_fun_nat_o(aTP_Lamm_lz(F),B))))),aa_int_int(divide_divide_int(groups3539618377306564664at_int(F,inf_inf_set_nat(A2,collect_nat(aa_int_fun_nat_o(aTP_Lamm_ma(F),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_8335_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_mb(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8336_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_mc(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_rat_rat(plus_plus_rat(aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8337_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_md(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8338_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_int,C: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_me(A),B),C),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_int_int(plus_plus_int(aa_complex_int(B,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8339_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_mf(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(B,A)),groups8097168146408367636l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups8097168146408367636l_real(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_8340_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_mg(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_rat_rat(plus_plus_rat(aa_real_rat(B,A)),groups1300246762558778688al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups1300246762558778688al_rat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_8341_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat,C: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_mh(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(B,A)),groups1935376822645274424al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups1935376822645274424al_nat(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_8342_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_int,C: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_mi(A),B),C),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_int_int(plus_plus_int(aa_real_int(B,A)),groups1932886352136224148al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))),groups1932886352136224148al_int(C,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_8343_sum_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: fun_o_real,C: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( groups8691415230153176458o_real(aa_fun688608109223240316o_real(aa_fun1133276423545935902o_real(aTP_Lamm_mj((A)),B),C),S2) = $ite(aa_set_o_o(member_o((A)),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_o_real(B,(A))),groups8691415230153176458o_real(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))),groups8691415230153176458o_real(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_8344_sum_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: fun_o_rat,C: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_mk((A)),B),C),S2) = $ite(aa_set_o_o(member_o((A)),S2),aa_rat_rat(plus_plus_rat(aa_o_rat(B,(A))),groups7872700643590313910_o_rat(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))),groups7872700643590313910_o_rat(C,minus_minus_set_o(S2,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_8345_tanh__altdef,axiom,
    ! [Xa: real] : aa_real_real(tanh_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xa)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xa)))) ).

% tanh_altdef
tff(fact_8346_tanh__altdef,axiom,
    ! [Xa: complex] : aa_complex_complex(tanh_complex,Xa) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Xa)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xa)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Xa)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xa)))) ).

% tanh_altdef
tff(fact_8347_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(Nb,A) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_8348_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_8349_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_8350_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8351_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( aa_set_int_o(member_int(B),minus_minus_set_int(B2,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,B))
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8352_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8353_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),B2)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8354_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( aa_set_int_o(member_int(B),minus_minus_set_int(B2,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,B))
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),B2)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8355_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B2)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8356_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,B))
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8357_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B2)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B2)
       => ( aa_set_int_o(member_int(B),minus_minus_set_int(B2,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,B))
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8358_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B2)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B2,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B))
           => ( ! [X3: complex] :
                  ( aa_set_complex_o(member_complex(X3),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8359_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B2)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B2)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B2,A2))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,B))
           => ( ! [X3: real] :
                  ( aa_set_real_o(member_real(X3),B2)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B2)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8360_member__le__sum,axiom,
    ! [Ia: complex,A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(member_complex(Ia),A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(A2,insert_complex(Ia,bot_bot_set_complex)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,Ia)),groups5808333547571424918x_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8361_member__le__sum,axiom,
    ! [Ia: real,A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(member_real(Ia),A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),minus_minus_set_real(A2,insert_real(Ia,bot_bot_set_real)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Ia)),groups8097168146408367636l_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8362_member__le__sum,axiom,
    ! [Ia: $o,A2: set_o,F: fun_o_real] :
      ( aa_set_o_o(member_o((Ia)),A2)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),minus_minus_set_o(A2,insert_o((Ia),bot_bot_set_o)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X3))) )
       => ( aa_set_o_o(finite_finite_o,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(Ia))),groups8691415230153176458o_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8363_member__le__sum,axiom,
    ! [Ia: int,A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(member_int(Ia),A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),minus_minus_set_int(A2,insert_int(Ia,bot_bot_set_int)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,Ia)),groups8778361861064173332t_real(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8364_member__le__sum,axiom,
    ! [Ia: complex,A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(member_complex(Ia),A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(A2,insert_complex(Ia,bot_bot_set_complex)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Ia)),groups5058264527183730370ex_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8365_member__le__sum,axiom,
    ! [Ia: real,A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(member_real(Ia),A2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),minus_minus_set_real(A2,insert_real(Ia,bot_bot_set_real)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
       => ( aa_set_real_o(finite_finite_real,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Ia)),groups1300246762558778688al_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8366_member__le__sum,axiom,
    ! [Ia: $o,A2: set_o,F: fun_o_rat] :
      ( aa_set_o_o(member_o((Ia)),A2)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),minus_minus_set_o(A2,insert_o((Ia),bot_bot_set_o)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(X3))) )
       => ( aa_set_o_o(finite_finite_o,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(Ia))),groups7872700643590313910_o_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8367_member__le__sum,axiom,
    ! [Ia: int,A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(member_int(Ia),A2)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),minus_minus_set_int(A2,insert_int(Ia,bot_bot_set_int)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
       => ( aa_set_int_o(finite_finite_int,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Ia)),groups3906332499630173760nt_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8368_member__le__sum,axiom,
    ! [Ia: nat,A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(Ia),A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(A2,insert_nat(Ia,bot_bot_set_nat)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Ia)),groups2906978787729119204at_rat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8369_member__le__sum,axiom,
    ! [Ia: complex,A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(member_complex(Ia),A2)
     => ( ! [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(A2,insert_complex(Ia,bot_bot_set_complex)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,Ia)),groups5693394587270226106ex_nat(F,A2)) ) ) ) ).

% member_le_sum
tff(fact_8370_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_8371_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_8372_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma)),Ma)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Ma) ) ).

% take_bit_nat_less_self_iff
tff(fact_8373_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_8374_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% mask_nat_less_exp
tff(fact_8375_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_8376_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_8377_convex__sum__bound__le,axiom,
    ! [I5: set_complex,Xa: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(Xa,I2)) )
     => ( ( groups6621422865394947399nteger(Xa,I5) = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_ml(Xa),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8378_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xa: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(Xa,I2)) )
     => ( ( groups7713935264441627589nteger(Xa,I5) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_mm(Xa),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8379_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xa: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xa,I2)) )
     => ( ( groups7501900531339628137nteger(Xa,I5) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_mn(Xa),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8380_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xa: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xa,I2)) )
     => ( ( groups7873554091576472773nteger(Xa,I5) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_mo(Xa),A),I5),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8381_convex__sum__bound__le,axiom,
    ! [I5: set_complex,Xa: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Xa,I2)) )
     => ( ( groups5808333547571424918x_real(Xa,I5) = one_one_real )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_mp(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8382_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xa: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Xa,I2)) )
     => ( ( groups8097168146408367636l_real(Xa,I5) = one_one_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_mq(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8383_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xa: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Xa,I2)) )
     => ( ( groups8778361861064173332t_real(Xa,I5) = one_one_real )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mr(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8384_convex__sum__bound__le,axiom,
    ! [I5: set_complex,Xa: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Xa,I2)) )
     => ( ( groups5058264527183730370ex_rat(Xa,I5) = one_one_rat )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_complex_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ms(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8385_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xa: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Xa,I2)) )
     => ( ( groups1300246762558778688al_rat(Xa,I5) = one_one_rat )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_mt(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8386_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xa: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xa,I2)) )
     => ( ( groups2906978787729119204at_rat(Xa,I5) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mu(Xa),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8387_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_8388_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_8389_exp__double,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_8390_exp__double,axiom,
    ! [Z: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_8391_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_8392_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_8393_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_8394_or__one__eq,axiom,
    ! [A: code_integer] : bit_se1080825931792720795nteger(A,one_one_Code_integer) = plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_8395_or__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(A),one_one_int) = aa_int_int(plus_plus_int(A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_8396_or__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_8397_one__or__eq,axiom,
    ! [A: code_integer] : bit_se1080825931792720795nteger(one_one_Code_integer,A) = plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_8398_one__or__eq,axiom,
    ! [A: int] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),A) = aa_int_int(plus_plus_int(A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_8399_one__or__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_8400_OR__upper,axiom,
    ! [Xa: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(Xa),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_8401_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_8402_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_8403_mask__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_8404_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_8405_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ).

% mask_half_int
tff(fact_8406_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) != aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(K),one_one_int)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_8407_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_8408_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_8409_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_8410_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_8411_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_8412_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K)),one_one_Code_integer) ).

% take_bit_numeral_minus_1_eq
tff(fact_8413_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_8414_take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_8415_take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_8416_take__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_8417_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_8418_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_8419_exp__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_8420_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_8421_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_8422_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_8423_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_8424_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( bit_se1745604003318907178nteger(Nb,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_8425_stable__imp__take__bit__eq,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_8426_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_8427_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_8428_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_8429_real__exp__bound__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa))) ) ) ).

% real_exp_bound_lemma
tff(fact_8430_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_8431_exp__lower__Taylor__quadratic,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xa)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_8432_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_8433_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_8434_or__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(1)
tff(fact_8435_or__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(5)
tff(fact_8436_log__base__10__eq1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xa)) ) ) ).

% log_base_10_eq1
tff(fact_8437_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_8438_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_8439_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_nat_numerals(4)
tff(fact_8440_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_8441_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_8442_log__less__cancel__iff,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_8443_log__less__one__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_8444_one__less__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_8445_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_8446_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_8447_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_8448_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% or_nat_numerals(3)
tff(fact_8449_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Ma: nat,Nb: nat] :
      groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Ma),zero_zero_complex,aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_8450_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Ma: nat,Nb: nat] :
      groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Ma),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_8451_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Ma: nat,Nb: nat] :
      groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Ma),zero_zero_int,aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_8452_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Ma: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Ma),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_8453_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Ma: nat,Nb: nat] :
      groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Ma),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_8454_log__le__cancel__iff,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_8455_log__le__one__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_8456_one__le__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_8457_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_8458_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_8459_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_8460_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_8461_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_8462_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_8463_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_8464_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_8465_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_8466_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_8467_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_8468_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_8469_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_8470_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_8471_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Ma)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_8472_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Ma),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_8473_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_8474_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_8475_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_8476_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_8477_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_8478_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_8479_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A2: set_nat] :
      groups2073611262835488442omplex(aTP_Lamm_mv(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_8480_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A2: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_mw(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_8481_sum__zero__power,axiom,
    ! [C: fun_nat_real,A2: set_nat] :
      groups6591440286371151544t_real(aTP_Lamm_mx(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_8482_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A2: set_nat] :
      groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_my(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)),
        zero_zero_complex ) ).

% sum_zero_power'
tff(fact_8483_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A2: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mz(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_rat_rat(divide_divide_rat(aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_8484_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A2: set_nat] :
      groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_na(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
        zero_zero_real ) ).

% sum_zero_power'
tff(fact_8485_or__minus__numerals_I4_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Ma),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_8486_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Ma: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Ma)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Ma),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_8487_or__minus__numerals_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Ma),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_8488_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Ma: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Ma)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Ma),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_8489_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_8490_num__induct,axiom,
    ! [P: fun_num_o,Xa: num] :
      ( aa_num_o(P,one)
     => ( ! [X3: num] :
            ( aa_num_o(P,X3)
           => aa_num_o(P,aa_num_num(inc,X3)) )
       => aa_num_o(P,Xa) ) ) ).

% num_induct
tff(fact_8491_add__inc,axiom,
    ! [Xa: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xa),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,Xa),Y)) ).

% add_inc
tff(fact_8492_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X3)),A2)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X3)) = aa_nat_nat(G,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8493_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X3)),A2)
           => ( aa_nat_real(F,aa_nat_nat(suc,X3)) = aa_nat_real(G,aa_nat_nat(suc,X3)) ) )
       => ( groups6591440286371151544t_real(F,A2) = groups6591440286371151544t_real(G,A2) ) ) ) ).

% sum_cong_Suc
tff(fact_8494_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Ma: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nb(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8495_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Ma: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb))) = groups6591440286371151544t_real(aTP_Lamm_nc(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8496_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Ma: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nd(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8497_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Ma: nat,K: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ne(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8498_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_8499_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Ma: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit1,Ma)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Ma)) ).

% or_not_num_neg.simps(6)
tff(fact_8500_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_8501_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Ma: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,Ma)) = aa_num_num(bit1,Ma) ).

% or_not_num_neg.simps(3)
tff(fact_8502_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_8503_inc_Osimps_I3_J,axiom,
    ! [Xa: num] : aa_num_num(inc,aa_num_num(bit1,Xa)) = aa_num_num(bit0,aa_num_num(inc,Xa)) ).

% inc.simps(3)
tff(fact_8504_inc_Osimps_I2_J,axiom,
    ! [Xa: num] : aa_num_num(inc,aa_num_num(bit0,Xa)) = aa_num_num(bit1,Xa) ).

% inc.simps(2)
tff(fact_8505_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Ma: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit0,Ma)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Ma)) ).

% or_not_num_neg.simps(5)
tff(fact_8506_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8507_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: complex] :
                ( aa_set_complex_o(member_complex(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8508_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: nat] :
                ( aa_set_nat_o(member_nat(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_8509_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,Nb) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) ) ) ).

% sum_SucD
tff(fact_8510_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,Ma: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit1,Ma)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Ma)) ).

% or_not_num_neg.simps(9)
tff(fact_8511_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_nat(F,X4) = one_one_nat )
            & ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8512_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_nat(F,X4) = one_one_nat )
            & ! [Xa4: complex] :
                ( aa_set_complex_o(member_complex(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8513_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_nat(F,X4) = one_one_nat )
            & ! [Xa4: nat] :
                ( aa_set_nat_o(member_nat(Xa4),A2)
               => ( ( X4 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_8514_add__One,axiom,
    ! [Xa: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xa),one) = aa_num_num(inc,Xa) ).

% add_One
tff(fact_8515_inc__BitM__eq,axiom,
    ! [Nb: num] : aa_num_num(inc,aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% inc_BitM_eq
tff(fact_8516_BitM__inc__eq,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(inc,Nb)) = aa_num_num(bit1,Nb) ).

% BitM_inc_eq
tff(fact_8517_mult__inc,axiom,
    ! [Xa: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xa),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,Xa),Y)),Xa) ).

% mult_inc
tff(fact_8518_sum__power__add,axiom,
    ! [Xa: complex,Ma: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_nf(Xa),Ma),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xa),Ma)),groups2073611262835488442omplex(power_power_complex(Xa),I5)) ).

% sum_power_add
tff(fact_8519_sum__power__add,axiom,
    ! [Xa: rat,Ma: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ng(Xa),Ma),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xa),Ma)),groups2906978787729119204at_rat(power_power_rat(Xa),I5)) ).

% sum_power_add
tff(fact_8520_sum__power__add,axiom,
    ! [Xa: int,Ma: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_nh(Xa),Ma),I5) = aa_int_int(times_times_int(aa_nat_int(power_power_int(Xa),Ma)),groups3539618377306564664at_int(power_power_int(Xa),I5)) ).

% sum_power_add
tff(fact_8521_sum__power__add,axiom,
    ! [Xa: real,Ma: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ni(Xa),Ma),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Ma)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xa),I5)) ).

% sum_power_add
tff(fact_8522_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Ma: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Ma)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_nj(G),Nb),Ma)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Ma)) ).

% sum.atLeastAtMost_rev
tff(fact_8523_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Ma: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Ma)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_nk(G),Nb),Ma),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Ma)) ).

% sum.atLeastAtMost_rev
tff(fact_8524_sum__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_cn,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ee(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8525_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_cn,collect_complex(aTP_Lamm_ec(Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8526_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Ma: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,Ma)) = aa_num_num(bit1,Ma) ).

% or_not_num_neg.simps(2)
tff(fact_8527_log__base__change,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_8528_sum__diff1__nat,axiom,
    ! [F: fun_complex_nat,A2: set_complex,A: complex] :
      groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),aa_complex_nat(F,A)),groups5693394587270226106ex_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_8529_sum__diff1__nat,axiom,
    ! [F: fun_set_nat_nat,A2: set_set_nat,A: set_nat] :
      groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A2,insert_set_nat(A,bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),aa_set_nat_nat(F,A)),groups8294997508430121362at_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_8530_sum__diff1__nat,axiom,
    ! [F: fun_real_nat,A2: set_real,A: real] :
      groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),aa_real_nat(F,A)),groups1935376822645274424al_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_8531_sum__diff1__nat,axiom,
    ! [F: fun_o_nat,A2: set_o,A: $o] :
      groups8507830703676809646_o_nat(F,minus_minus_set_o(A2,insert_o((A),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(F,A2)),aa_o_nat(F,(A))),groups8507830703676809646_o_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_8532_sum__diff1__nat,axiom,
    ! [F: fun_int_nat,A2: set_int,A: int] :
      groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)),groups4541462559716669496nt_nat(F,A2)) ).

% sum_diff1_nat
tff(fact_8533_sum__diff1__nat,axiom,
    ! [F: fun_nat_nat,A2: set_nat,A: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_nat_nat(F,A)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) ).

% sum_diff1_nat
tff(fact_8534_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8535_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8536_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8537_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8538_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8539_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8540_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8541_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8542_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8543_sum_OatLeast__Suc__atMost,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Ma)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8544_sum_OatLeast__Suc__atMost,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = aa_int_int(plus_plus_int(aa_nat_int(G,Ma)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8545_sum_OatLeast__Suc__atMost,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Ma)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8546_sum_OatLeast__Suc__atMost,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Ma)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8547_sum_Onat__ivl__Suc_H,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8548_sum_Onat__ivl__Suc_H,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = aa_int_int(plus_plus_int(aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8549_sum_Onat__ivl__Suc_H,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8550_sum_Onat__ivl__Suc_H,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8551_numeral__inc,axiom,
    ! [Xa: num] : aa_num_Extended_enat(numera1916890842035813515d_enat,aa_num_num(inc,Xa)) = plus_p3455044024723400733d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Xa),one_on7984719198319812577d_enat) ).

% numeral_inc
tff(fact_8552_numeral__inc,axiom,
    ! [Xa: num] : numera6690914467698888265omplex(aa_num_num(inc,Xa)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xa)),one_one_complex) ).

% numeral_inc
tff(fact_8553_numeral__inc,axiom,
    ! [Xa: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xa)),one_one_real) ).

% numeral_inc
tff(fact_8554_numeral__inc,axiom,
    ! [Xa: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xa)),one_one_nat) ).

% numeral_inc
tff(fact_8555_numeral__inc,axiom,
    ! [Xa: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,Xa)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Xa)),one_one_int) ).

% numeral_inc
tff(fact_8556_numeral__inc,axiom,
    ! [Xa: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Xa)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xa)),one_one_rat) ).

% numeral_inc
tff(fact_8557_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Ma: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit0,Ma)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Ma)) ).

% or_not_num_neg.simps(8)
tff(fact_8558_sum_OSuc__reindex__ivl,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Ma)),groups2906978787729119204at_rat(aTP_Lamm_nl(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8559_sum_OSuc__reindex__ivl,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(plus_plus_int(aa_nat_int(G,Ma)),groups3539618377306564664at_int(aTP_Lamm_nm(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8560_sum_OSuc__reindex__ivl,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Ma)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nb(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8561_sum_OSuc__reindex__ivl,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Ma)),groups6591440286371151544t_real(aTP_Lamm_nc(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8562_sum__Suc__diff,axiom,
    ! [Ma: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(aTP_Lamm_nn(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Nb))),aa_nat_rat(F,Ma)) ) ) ).

% sum_Suc_diff
tff(fact_8563_sum__Suc__diff,axiom,
    ! [Ma: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(aTP_Lamm_no(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = aa_int_int(minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Ma)) ) ) ).

% sum_Suc_diff
tff(fact_8564_sum__Suc__diff,axiom,
    ! [Ma: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( groups6591440286371151544t_real(aTP_Lamm_np(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Ma)) ) ) ).

% sum_Suc_diff
tff(fact_8565_sum__Un__nat,axiom,
    ! [A2: set_int,B2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(finite_finite_int,B2)
       => ( groups4541462559716669496nt_nat(F,sup_sup_set_int(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2))),groups4541462559716669496nt_nat(F,inf_inf_set_int(A2,B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8566_sum__Un__nat,axiom,
    ! [A2: set_complex,B2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B2)
       => ( groups5693394587270226106ex_nat(F,sup_sup_set_complex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2))),groups5693394587270226106ex_nat(F,inf_inf_set_complex(A2,B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8567_sum__Un__nat,axiom,
    ! [A2: set_nat,B2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),sup_sup_set_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2))),aa_set_nat_nat(groups3542108847815614940at_nat(F),inf_inf_set_nat(A2,B2))) ) ) ) ).

% sum_Un_nat
tff(fact_8568_log__mult,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_8569_log__divide,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_8570_sum_Oub__add__nat,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8571_sum_Oub__add__nat,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(plus_plus_int(groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8572_sum_Oub__add__nat,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8573_sum_Oub__add__nat,axiom,
    ! [Ma: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8574_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_8575_or__not__num__neg_Oelims,axiom,
    ! [Xa: num,Xaa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xa),Xaa) = Y )
     => ( ( ( Xa = one )
         => ( ( Xaa = one )
           => ( Y != one ) ) )
       => ( ( ( Xa = one )
           => ! [M2: num] :
                ( ( Xaa = aa_num_num(bit0,M2) )
               => ( Y != aa_num_num(bit1,M2) ) ) )
         => ( ( ( Xa = one )
             => ! [M2: num] :
                  ( ( Xaa = aa_num_num(bit1,M2) )
                 => ( Y != aa_num_num(bit1,M2) ) ) )
           => ( ( ? [N: num] : Xa = aa_num_num(bit0,N)
               => ( ( Xaa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( Xa = aa_num_num(bit0,N) )
                   => ! [M2: num] :
                        ( ( Xaa = aa_num_num(bit0,M2) )
                       => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
               => ( ! [N: num] :
                      ( ( Xa = aa_num_num(bit0,N) )
                     => ! [M2: num] :
                          ( ( Xaa = aa_num_num(bit1,M2) )
                         => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
                 => ( ( ? [N: num] : Xa = aa_num_num(bit1,N)
                     => ( ( Xaa = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( Xa = aa_num_num(bit1,N) )
                         => ! [M2: num] :
                              ( ( Xaa = aa_num_num(bit0,M2) )
                             => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xa = aa_num_num(bit1,N) )
                           => ! [M2: num] :
                                ( ( Xaa = aa_num_num(bit1,M2) )
                               => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_8576_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8577_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,Ma: nat,Nb: nat] :
      groups2073611262835488442omplex(aTP_Lamm_nq(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb),aa_complex_complex(minus_minus_complex(aa_nat_complex(F,Ma)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_8578_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,Ma: nat,Nb: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_nr(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb),aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Ma)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_8579_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Ma: nat,Nb: nat] :
      groups3539618377306564664at_int(aTP_Lamm_ns(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb),aa_int_int(minus_minus_int(aa_nat_int(F,Ma)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_8580_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Ma: nat,Nb: nat] :
      groups6591440286371151544t_real(aTP_Lamm_nt(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Ma)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_8581_sum__telescope_H_H,axiom,
    ! [Ma: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_nu(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Nb)),aa_nat_rat(F,Ma)) ) ) ).

% sum_telescope''
tff(fact_8582_sum__telescope_H_H,axiom,
    ! [Ma: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_nv(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb)) = aa_int_int(minus_minus_int(aa_nat_int(F,Nb)),aa_nat_int(F,Ma)) ) ) ).

% sum_telescope''
tff(fact_8583_sum__telescope_H_H,axiom,
    ! [Ma: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( groups6591440286371151544t_real(aTP_Lamm_nw(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma)),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Nb)),aa_nat_real(F,Ma)) ) ) ).

% sum_telescope''
tff(fact_8584_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ea,Nb))) ).

% mask_eq_sum_exp
tff(fact_8585_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ea,Nb))) ).

% mask_eq_sum_exp
tff(fact_8586_sum__gp__multiplied,axiom,
    ! [Ma: nat,Nb: nat,Xa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xa)),groups2073611262835488442omplex(power_power_complex(Xa),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xa),Ma)),aa_nat_complex(power_power_complex(Xa),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8587_sum__gp__multiplied,axiom,
    ! [Ma: nat,Nb: nat,Xa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xa)),groups2906978787729119204at_rat(power_power_rat(Xa),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xa),Ma)),aa_nat_rat(power_power_rat(Xa),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8588_sum__gp__multiplied,axiom,
    ! [Ma: nat,Nb: nat,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_int_int(times_times_int(aa_int_int(minus_minus_int(one_one_int),Xa)),groups3539618377306564664at_int(power_power_int(Xa),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(Xa),Ma)),aa_nat_int(power_power_int(Xa),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8589_sum__gp__multiplied,axiom,
    ! [Ma: nat,Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xa)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_8590_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Ma: nat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_nx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ).

% sum.in_pairs
tff(fact_8591_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Ma: nat,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_ny(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ).

% sum.in_pairs
tff(fact_8592_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Ma: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ).

% sum.in_pairs
tff(fact_8593_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Ma: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups6591440286371151544t_real(aTP_Lamm_oa(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) ).

% sum.in_pairs
tff(fact_8594_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_or_eq
tff(fact_8595_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% or_Suc_0_eq
tff(fact_8596_or__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_8597_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ea,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_8598_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cq),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8599_or__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Ma),Nb) = $ite(
        Ma = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Ma,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,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_8600_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ob(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8601_Sum__Icc__nat,axiom,
    ! [Ma: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cq),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_8602_log__base__10__eq2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xa)) ) ) ).

% log_base_10_eq2
tff(fact_8603_sum__gp,axiom,
    ! [Xa: rat,Ma: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xa),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma),
        zero_zero_rat,
        $ite(Xa = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Ma)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xa),Ma)),aa_nat_rat(power_power_rat(Xa),aa_nat_nat(suc,Nb)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xa))) ) ).

% sum_gp
tff(fact_8604_sum__gp,axiom,
    ! [Xa: complex,Ma: nat,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(Xa),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma),
        zero_zero_complex,
        $ite(Xa = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Ma)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xa),Ma)),aa_nat_complex(power_power_complex(Xa),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xa))) ) ).

% sum_gp
tff(fact_8605_sum__gp,axiom,
    ! [Xa: real,Ma: nat,Nb: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_set_nat(set_or1269000886237332187st_nat(Ma),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma),
        zero_zero_real,
        $ite(Xa = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Ma)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(suc,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xa))) ) ).

% sum_gp
tff(fact_8606_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_8607_log2__of__power__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_8608_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8609_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8610_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_8611_arctan__half,axiom,
    ! [Xa: real] : aa_real_real(arctan,Xa) = 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(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_8612_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Ma) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_8613_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ma) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_8614_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Ma) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_8615_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Ma) = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_8616_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Ma) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_8617_int__eq__iff__numeral,axiom,
    ! [Ma: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ma) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Ma = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_8618_bit__0__eq,axiom,
    bit_se1146084159140164899it_int(zero_zero_int) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_8619_bit__0__eq,axiom,
    aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,zero_zero_nat) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_8620_abs__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_8621_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_8622_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_8623_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_8624_negative__eq__positive,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Ma) )
    <=> ( ( Nb = zero_zero_nat )
        & ( Ma = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_8625_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sqrt,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_8626_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_8627_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Ma) = zero_zero_rat )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_8628_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Ma) = zero_zero_real )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_8629_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ma) = zero_zero_int )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_8630_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Ma) = zero_zero_nat )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_8631_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Ma) = zero_zero_complex )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_8632_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Ma) = zero_z3403309356797280102nteger )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_8633_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_8634_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_8635_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_8636_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_8637_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_8638_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_8639_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_8640_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_8641_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_8642_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_8643_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_8644_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_8645_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_8646_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_8647_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_8648_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_8649_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_8650_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_8651_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_Extended_enat(numera1916890842035813515d_enat,Nb) ).

% of_nat_numeral
tff(fact_8652_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% of_nat_numeral
tff(fact_8653_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% of_nat_numeral
tff(fact_8654_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_8655_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% of_nat_numeral
tff(fact_8656_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_8657_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,Nb)) = numera6620942414471956472nteger(Nb) ).

% of_nat_numeral
tff(fact_8658_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_8659_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_8660_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_8661_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_8662_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_8663_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_8664_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_8665_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_8666_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_8667_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Ma)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_add
tff(fact_8668_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Ma),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_add
tff(fact_8669_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_8670_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_8671_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_8672_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_8673_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Ma)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_mult
tff(fact_8674_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mult
tff(fact_8675_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_8676_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_8677_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_8678_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_8679_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_8680_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = one_one_Code_integer )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_8681_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_8682_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_8683_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_8684_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_8685_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_8686_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_8687_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_8688_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_8689_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_8690_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_8691_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_8692_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_8693_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xa) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_8694_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xa) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_8695_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xa) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_8696_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xa) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_8697_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Xa) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) )
    <=> ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_8698_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_8699_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_8700_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_8701_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_8702_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,Xa) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xa ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_8703_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),Nb) ).

% of_nat_power
tff(fact_8704_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Ma)),Nb) ).

% of_nat_power
tff(fact_8705_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),Nb) ).

% of_nat_power
tff(fact_8706_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,Ma)),Nb) ).

% of_nat_power
tff(fact_8707_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),Nb) ).

% of_nat_power
tff(fact_8708_negative__zless,axiom,
    ! [Nb: nat,Ma: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Ma)) ).

% negative_zless
tff(fact_8709_real__sqrt__lt__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_8710_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_8711_real__sqrt__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_8712_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_8713_real__sqrt__abs2,axiom,
    ! [Xa: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)) = aa_real_real(abs_abs_real,Xa) ).

% real_sqrt_abs2
tff(fact_8714_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_8715_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_real(zero_n3304061248610475627l_real,(P)) ).

% of_nat_of_bool
tff(fact_8716_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_complex(semiri8010041392384452111omplex,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_complex(zero_n1201886186963655149omplex,(P)) ).

% of_nat_of_bool
tff(fact_8717_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% of_nat_of_bool
tff(fact_8718_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_8719_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n356916108424825756nteger((P)) ).

% of_nat_of_bool
tff(fact_8720_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),zero_zero_real)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_8721_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),zero_z3403309356797280102nteger)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_8722_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),zero_zero_rat)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_8723_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),zero_zero_nat)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_8724_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),zero_zero_int)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_8725_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Ma)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Ma)) ).

% of_nat_Suc
tff(fact_8726_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Ma)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Ma)) ).

% of_nat_Suc
tff(fact_8727_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Ma)) = aa_int_int(plus_plus_int(one_one_int),aa_nat_int(semiri1314217659103216013at_int,Ma)) ).

% of_nat_Suc
tff(fact_8728_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Ma)) ).

% of_nat_Suc
tff(fact_8729_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Ma)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Ma)) ).

% of_nat_Suc
tff(fact_8730_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Ma)) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)) ).

% of_nat_Suc
tff(fact_8731_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Ma)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_8732_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_8733_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Ma)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_8734_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_8735_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_8736_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_8737_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),zero_zero_int)
    <=> aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_8738_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_8739_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_Extended_enat_o(aa_Ext6368937152127170207enat_o(ord_le72135733267957522d_enat,zero_z5237406670263579293d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_8740_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_8741_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_8742_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_8743_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_8744_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_8745_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_8746_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_8747_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_8748_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_8749_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_8750_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_8751_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_8752_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_8753_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_8754_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Nb: nat] :
      ( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y) = aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Xa)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_8755_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_8756_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_8757_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_8758_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_8759_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xa)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_8760_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xa: num,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xa)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_8761_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: nat] :
      ( ( aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_num_Extended_enat(numera1916890842035813515d_enat,Xa)),Nb) = aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_8762_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xa)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_8763_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xa)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_8764_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_8765_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_8766_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xa)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_8767_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xa)),Nb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_8768_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_8769_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_8770_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_8771_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_8772_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_8773_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_8774_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_8775_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_8776_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_8777_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_8778_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_num_real(numeral_numeral_real,W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_8779_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_8780_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Ma))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Ma) ) ).

% numeral_le_real_of_nat_iff
tff(fact_8781_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(2)
tff(fact_8782_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(2)
tff(fact_8783_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_8784_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(3)
tff(fact_8785_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(3)
tff(fact_8786_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_8787_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xa)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_8788_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_8789_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xa)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_8790_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_8791_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_8792_bit__0,axiom,
    ! [A: code_integer] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),zero_zero_nat)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_8793_bit__0,axiom,
    ! [A: int] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),zero_zero_nat)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_8794_bit__0,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),zero_zero_nat)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_8795_real__sqrt__abs,axiom,
    ! [Xa: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xa) ).

% real_sqrt_abs
tff(fact_8796_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_8797_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_8798_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_8799_real__sqrt__pow2__iff,axiom,
    ! [Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% real_sqrt_pow2_iff
tff(fact_8800_real__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa ) ) ).

% real_sqrt_pow2
tff(fact_8801_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xa: real,Y: real,Xaa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_8802_real__sqrt__mult,axiom,
    ! [Xa: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_8803_bit__and__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se725231765392027082nd_int(K),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_8804_bit__or__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se1409905431419307370or_int(K),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        | aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_8805_real__sqrt__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_gt_zero
tff(fact_8806_real__sqrt__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_ge_zero
tff(fact_8807_real__sqrt__eq__zero__cancel,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(sqrt,Xa) = zero_zero_real )
       => ( Xa = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_8808_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_8809_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_8810_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_8811_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_8812_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_8813_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_8814_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_8815_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_8816_zadd__int__left,axiom,
    ! [Ma: nat,Nb: nat,Z: int] : aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))),Z) ).

% zadd_int_left
tff(fact_8817_int__plus,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Ma)) ).

% int_plus
tff(fact_8818_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_8819_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(times_times_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_8820_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_8821_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_8822_zmod__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_8823_nat__less__as__int,axiom,
    ! [X: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_less_as_int
tff(fact_8824_nat__leq__as__int,axiom,
    ! [X: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_leq_as_int
tff(fact_8825_real__div__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_real_real(sqrt,Xa)) = aa_real_real(sqrt,Xa) ) ) ).

% real_div_sqrt
tff(fact_8826_sqrt__add__le__add__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_8827_le__real__sqrt__sumsq,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_8828_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% bit_not_int_iff'
tff(fact_8829_reals__Archimedean3,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ! [Y4: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xa)) ) ).

% reals_Archimedean3
tff(fact_8830_int__cases4,axiom,
    ! [Ma: int] :
      ( ! [N: nat] : Ma != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( Ma != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_8831_real__of__nat__div4,axiom,
    ! [Nb: nat,Xa: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xa))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xa))) ).

% real_of_nat_div4
tff(fact_8832_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_8833_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_8834_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N4: nat] : Z = aa_int_int(plus_plus_int(W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_8835_int__zle__neg,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Ma)))
    <=> ( ( Nb = zero_zero_nat )
        & ( Ma = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_8836_real__of__nat__div,axiom,
    ! [D: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_8837_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_8838_negative__zle__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),zero_zero_int) ).

% negative_zle_0
tff(fact_8839_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_8840_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Ma: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Ma),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_8841_bit__concat__bit__iff,axiom,
    ! [Ma: nat,K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(Ma,K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
          & aa_nat_o(bit_se1146084159140164899it_int(K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_8842_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_8843_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_8844_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_8845_nat__less__real__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Ma)) ) ).

% nat_less_real_le
tff(fact_8846_nat__le__real__less,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_8847_zmult__zless__mono2__lemma,axiom,
    ! [Ia: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),Ia)),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_8848_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_8849_negD,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
     => ? [N: nat] : Xa = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_8850_negative__zless__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_8851_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_8852_real__of__nat__div__aux,axiom,
    ! [Xa: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_8853_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(bit_concat_bit(Nb,K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_8854_real__less__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_8855_real__le__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_8856_sqrt__le__D,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_8857_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),M)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),N) ) ) ) ).

% int_bit_bound
tff(fact_8858_real__archimedian__rdiv__eq__0,axiom,
    ! [Xa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),Xa)),C) )
         => ( Xa = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_8859_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_8860_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xa: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
         => aa_int_o(P,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_8861_real__of__nat__div2,axiom,
    ! [Nb: nat,Xa: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xa))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xa)))) ).

% real_of_nat_div2
tff(fact_8862_real__of__nat__div3,axiom,
    ! [Nb: nat,Xa: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xa))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xa)))),one_one_real) ).

% real_of_nat_div3
tff(fact_8863_log__base__pow,axiom,
    ! [A: real,Nb: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_8864_ln__realpow,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_realpow
tff(fact_8865_log__nat__power,axiom,
    ! [Xa: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) ) ) ).

% log_nat_power
tff(fact_8866_real__sqrt__unique,axiom,
    ! [Y: real,Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,Xa) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_8867_real__le__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_8868_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_8869_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( Xa = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_8870_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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))))) = Xa )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_8871_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_8872_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_8873_real__sqrt__sum__squares__ge1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_8874_sqrt__ge__absD,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_8875_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_int_def
tff(fact_8876_log2__of__power__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma)) ) ) ).

% log2_of_power_eq
tff(fact_8877_linear__plus__1__le__power,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xa)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_8878_log__of__power__less,axiom,
    ! [Ma: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_8879_Bernoulli__inequality,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_8880_real__less__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_8881_sqrt__sum__squares__le__sum,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_8882_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_8883_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_8884_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_8885_real__sqrt__ge__abs1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_8886_ln__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xa)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_8887_arsinh__real__def,axiom,
    ! [Xa: real] : aa_real_real(arsinh_real,Xa) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_8888_log__of__power__le,axiom,
    ! [Ma: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_8889_arsinh__real__aux,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_8890_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_8891_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xa: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_8892_arith__geo__mean__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_8893_less__log2__of__power,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Ma)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))) ) ).

% less_log2_of_power
tff(fact_8894_le__log2__of__power,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Ma)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))) ) ).

% le_log2_of_power
tff(fact_8895_set__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(Nb),K) = aa_int_int(plus_plus_int(K),aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% set_bit_eq
tff(fact_8896_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(Nb),K) = aa_int_int(minus_minus_int(K),aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% unset_bit_eq
tff(fact_8897_cos__x__y__le__one,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_8898_real__sqrt__sum__squares__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_8899_arcosh__real__def,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_real(arcosh_real,Xa) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_8900_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_8901_log2__of__power__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_8902_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_8903_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xa)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_8904_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xa))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_8905_sqrt__sum__squares__half__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_8906_monoseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_oc,Xa)) ) ).

% monoseq_arctan_series
tff(fact_8907_ln__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xa) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_od,Xa)) ) ) ) ).

% ln_series
tff(fact_8908_arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(arctan,Xa) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_oe,Xa)) ) ) ).

% arctan_series
tff(fact_8909_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,
        $ite((P),A,B)) = $ite((P),aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_if
tff(fact_8910_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_8911_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_8912_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_8913_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_8914_bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Ma),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_8915_monoseq__realpow,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xa)) ) ) ).

% monoseq_realpow
tff(fact_8916_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_of) ).

% pi_series
tff(fact_8917_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8918_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_8919_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_8920_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_8921_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_8922_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_8923_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8924_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_8925_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_8926_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_8927_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_8928_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_8929_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_8930_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_8931_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_8932_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_8933_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_8934_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_8935_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_8936_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_8937_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_8938_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_8939_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_og(F),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_oh(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),groups6591440286371151544t_real(aTP_Lamm_oi(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_8940_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_8941_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_8942_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_8943_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_oj,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8944_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_8945_summable__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_oe,Xa)) ) ).

% summable_arctan_series
tff(fact_8946_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_8947_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8948_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_8949_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_8950_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_8951_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_8952_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_8953_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_8954_sin__periodic,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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,Xa) ).

% sin_periodic
tff(fact_8955_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_8956_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_8957_sin__2pi__minus,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xa)) ).

% sin_2pi_minus
tff(fact_8958_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8959_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,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_8960_sin__x__le__x,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),Xa) ) ).

% sin_x_le_x
tff(fact_8961_sin__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero
tff(fact_8962_sin__x__ge__neg__x,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xa)),aa_real_real(sin_real,Xa)) ) ).

% sin_x_ge_neg_x
tff(fact_8963_sin__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_ge_zero
tff(fact_8964_sin__eq__0__pi,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),pi)
       => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
         => ( Xa = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8965_sin__zero__pi__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),pi)
     => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8966_sin__zero__iff__int2,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
    <=> ? [I: int] : Xa = 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_8967_sin__gt__zero__02,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero_02
tff(fact_8968_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8969_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ok(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8970_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_8971_sin__gt__zero2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero2
tff(fact_8972_sin__lt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xa)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8973_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_8974_sin__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xa) = aa_real_real(sin_real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_8975_sin__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8976_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xa)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8977_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_8978_sin__le__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_8979_sin__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xa)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_8980_sin__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8981_sin__monotone__2pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xa)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8982_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X3) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_8983_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8984_sin__arctan,axiom,
    ! [Xa: real] : aa_real_real(sin_real,aa_real_real(arctan,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_8985_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D3)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D3)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_8986_sin__zero__iff__int,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
    <=> ? [I: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I)
          & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_8987_sin__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_8988_sin__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xa = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_8989_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_8990_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_8991_cos__pi__eq__zero,axiom,
    ! [Ma: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8992_sincos__total__2pi,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xa = aa_real_real(cos_real,T5) )
               => ( Y != aa_real_real(sin_real,T5) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8993_sin__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xa)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8994_Maclaurin__exp__lt,axiom,
    ! [Xa: real,Nb: nat] :
      ( ( Xa != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(exp_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ol,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8995_ceiling__log__eq__powr__iff,axiom,
    ! [Xa: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8996_powr__gt__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A))
    <=> ( Xa != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_8997_powr__nonneg__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xa)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_8998_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_8999_powr__eq__one__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xa) = one_one_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_9000_powr__one__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),one_one_real) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% powr_one_gt_zero_iff
tff(fact_9001_powr__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),one_one_real) = Xa ) ) ).

% powr_one
tff(fact_9002_numeral__powr__numeral__real,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_9003_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_9004_powr__log__cancel,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)) = Xa ) ) ) ) ).

% powr_log_cancel
tff(fact_9005_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_9006_tan__periodic__n,axiom,
    ! [Xa: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,Xa) ).

% tan_periodic_n
tff(fact_9007_tan__periodic__nat,axiom,
    ! [Xa: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xa) ).

% tan_periodic_nat
tff(fact_9008_tan__periodic__int,axiom,
    ! [Xa: real,Ia: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Ia)),pi))) = aa_real_real(tan_real,Xa) ).

% tan_periodic_int
tff(fact_9009_powr__numeral,axiom,
    ! [Xa: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_9010_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_9011_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_9012_cos__periodic,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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,Xa) ).

% cos_periodic
tff(fact_9013_cos__2pi__minus,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xa)) = aa_real_real(cos_real,Xa) ).

% cos_2pi_minus
tff(fact_9014_tan__periodic,axiom,
    ! [Xa: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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,Xa) ).

% tan_periodic
tff(fact_9015_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi2
tff(fact_9016_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi
tff(fact_9017_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_9018_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_9019_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_9020_square__powr__half,axiom,
    ! [Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xa) ).

% square_powr_half
tff(fact_9021_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_9022_powr__powr,axiom,
    ! [Xa: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% powr_powr
tff(fact_9023_powr__non__neg,axiom,
    ! [A: real,Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xa)),zero_zero_real) ).

% powr_non_neg
tff(fact_9024_powr__less__mono2__neg,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_9025_powr__mono2,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mono2
tff(fact_9026_powr__ge__pzero,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),Y)) ).

% powr_ge_pzero
tff(fact_9027_polar__Ex,axiom,
    ! [Xa: real,Y: real] :
    ? [R: real,A4: real] :
      ( ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(cos_real,A4)) )
      & ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(sin_real,A4)) ) ) ).

% polar_Ex
tff(fact_9028_cos__arctan__not__zero,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(arctan,Xa)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_9029_powr__mono2_H,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)) ) ) ) ).

% powr_mono2'
tff(fact_9030_powr__less__mono2,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_less_mono2
tff(fact_9031_powr__inj,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xa) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( Xa = Y ) ) ) ) ).

% powr_inj
tff(fact_9032_gr__one__powr,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),Y)) ) ) ).

% gr_one_powr
tff(fact_9033_ge__one__powr__ge__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_9034_powr__mono__both,axiom,
    ! [A: real,B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_9035_powr__le1,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_9036_powr__divide,axiom,
    ! [Xa: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_divide
tff(fact_9037_powr__mult,axiom,
    ! [Xa: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_9038_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_9039_log__base__powr,axiom,
    ! [A: real,B: real,Xa: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B)),Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)),B) ) ) ).

% log_base_powr
tff(fact_9040_ln__powr,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_powr
tff(fact_9041_log__powr,axiom,
    ! [Xa: real,B: real,Y: real] :
      ( ( Xa != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) ) ) ).

% log_powr
tff(fact_9042_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_9043_cos__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_9044_cos__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,Xa) = aa_real_real(cos_real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_9045_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_9046_powr__realpow,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb) ) ) ).

% powr_realpow
tff(fact_9047_less__log__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xa) ) ) ) ).

% less_log_iff
tff(fact_9048_log__less__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_less_iff
tff(fact_9049_less__powr__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)),Y) ) ) ) ).

% less_powr_iff
tff(fact_9050_powr__less__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xa)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) ) ) ) ).

% powr_less_iff
tff(fact_9051_cos__monotone__0__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_9052_cos__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_9053_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_9054_sin__zero__abs__cos__one,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xa)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_9055_powr__neg__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xa) ) ) ).

% powr_neg_one
tff(fact_9056_powr__mult__base,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_9057_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_9058_powr__le__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xa)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) ) ) ) ).

% powr_le_iff
tff(fact_9059_le__powr__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)),Y) ) ) ) ).

% le_powr_iff
tff(fact_9060_log__le__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_le_iff
tff(fact_9061_le__log__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xa) ) ) ) ).

% le_log_iff
tff(fact_9062_cos__is__zero,axiom,
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X3) = zero_zero_real )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X3 ) ) ) ).

% cos_is_zero
tff(fact_9063_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_9064_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_9065_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),pi)
            & ( aa_real_real(cos_real,X3) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),pi)
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_9066_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_9067_cos__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_9068_ln__powr__bound,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_9069_ln__powr__bound2,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xa)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xa)) ) ) ).

% ln_powr_bound2
tff(fact_9070_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_9071_add__log__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xa)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_9072_log__add__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_9073_minus__log__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xa)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_9074_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_9075_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_9076_sin__cos__le1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_9077_tan__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xa)) ) ) ).

% tan_gt_zero
tff(fact_9078_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X3)) ) ) ).

% lemma_tan_total
tff(fact_9079_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X3) = Y ) ) ).

% lemma_tan_total1
tff(fact_9080_tan__mono__lt__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_9081_tan__monotone_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xa)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_9082_tan__monotone,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xa)) ) ) ) ).

% tan_monotone
tff(fact_9083_tan__total,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X3) = Y )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X3 ) ) ) ).

% tan_total
tff(fact_9084_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_9085_cos__double__less__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_9086_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_9087_cos__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xa)) ) ) ).

% cos_gt_zero
tff(fact_9088_log__minus__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_9089_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_9090_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_9091_cos__one__2pi__int,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = one_one_real )
    <=> ? [X4: int] : Xa = 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_9092_Maclaurin__cos__expansion,axiom,
    ! [Xa: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_om,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9093_powr__half__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xa) ) ) ).

% powr_half_sqrt
tff(fact_9094_powr__neg__numeral,axiom,
    ! [Xa: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_9095_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X3) = Y ) ) ) ).

% tan_total_pos
tff(fact_9096_tan__pos__pi2__le,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xa)) ) ) ).

% tan_pos_pi2_le
tff(fact_9097_tan__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xa)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_9098_tan__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_9099_tan__mono__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_9100_tan__bound__pi2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xa))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_9101_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_9102_cos__gt__zero__pi,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xa)) ) ) ).

% cos_gt_zero_pi
tff(fact_9103_cos__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xa)) ) ) ).

% cos_ge_zero
tff(fact_9104_arctan__unique,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xa) = Y )
         => ( aa_real_real(arctan,Y) = Xa ) ) ) ) ).

% arctan_unique
tff(fact_9105_arctan__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xa)) = Xa ) ) ) ).

% arctan_tan
tff(fact_9106_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_9107_cos__one__2pi,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = one_one_real )
    <=> ( ? [X4: nat] : Xa = 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] : Xa = 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_9108_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_on(H),J),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_9109_Maclaurin__cos__expansion2,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xa)
            & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_om,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9110_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),zero_zero_real)
            & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_om,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9111_tan__total__pi4,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ? [Z4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z4)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z4) = Xa ) ) ) ).

% tan_total_pi4
tff(fact_9112_cos__arctan,axiom,
    ! [Xa: real] : aa_real_real(cos_real,aa_real_real(arctan,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_9113_Maclaurin__exp__le,axiom,
    ! [Xa: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(exp_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ol,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_9114_sincos__total__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),pi)
            & ( Xa = aa_real_real(cos_real,T5) )
            & ( Y = aa_real_real(sin_real,T5) ) ) ) ) ).

% sincos_total_pi
tff(fact_9115_sin__cos__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xa))
     => ( aa_real_real(sin_real,Xa) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_9116_sin__expansion__lemma,axiom,
    ! [Xa: real,Ma: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Ma))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_9117_cos__zero__iff__int,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = zero_zero_real )
    <=> ? [I: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I)
          & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_9118_cos__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(cos_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X)),zero_zero_real) ).

% cos_coeff_def
tff(fact_9119_cos__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(cos_real,Xa) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_9120_cos__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(cos_real,Xa) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xa = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_9121_cos__expansion__lemma,axiom,
    ! [Xa: real,Ma: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Ma))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_9122_sincos__total__pi__half,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xa = aa_real_real(cos_real,T5) )
              & ( Y = aa_real_real(sin_real,T5) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_9123_sincos__total__2pi__le,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),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] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xa = aa_real_real(cos_real,T5) )
          & ( Y = aa_real_real(sin_real,T5) ) ) ) ).

% sincos_total_2pi_le
tff(fact_9124_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xa)
            & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oo,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9125_Maclaurin__sin__expansion4,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),Xa)
          & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oo,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9126_Maclaurin__sin__expansion2,axiom,
    ! [Xa: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oo,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9127_Maclaurin__sin__expansion,axiom,
    ! [Xa: real,Nb: nat] :
    ? [T5: real] : aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oo,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_9128_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_9129_fact__less__mono__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Ma)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_9130_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_9131_fact__diff__Suc,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_9132_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_9133_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_9134_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_9135_sin__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(sin_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X))) ).

% sin_coeff_def
tff(fact_9136_sin__paired,axiom,
    ! [Xa: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_op,Xa)),aa_real_real(sin_real,Xa)) ).

% sin_paired
tff(fact_9137_cos__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xa)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_9138_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_9139_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_9140_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_9141_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_9142_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_9143_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_9144_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_9145_arccos__cos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xa)) = Xa ) ) ) ).

% arccos_cos
tff(fact_9146_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_9147_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_9148_sin__arccos__nonzero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xa)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9149_arccos__cos2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xa)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xa)) = aa_real_real(uminus_uminus_real,Xa) ) ) ) ).

% arccos_cos2
tff(fact_9150_cos__arcsin__nonzero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xa)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9151_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_oq),one_one_real) ).

% power_half_series
tff(fact_9152_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_9153_sums__if_H,axiom,
    ! [G: fun_nat_real,Xa: real] :
      ( aa_real_o(sums_real(G),Xa)
     => aa_real_o(sums_real(aTP_Lamm_or(G)),Xa) ) ).

% sums_if'
tff(fact_9154_sums__if,axiom,
    ! [G: fun_nat_real,Xa: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),Xa)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_os(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) ) ) ).

% sums_if
tff(fact_9155_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_9156_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9157_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_9158_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_9159_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_9160_arcsin__sin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xa)) = Xa ) ) ) ).

% arcsin_sin
tff(fact_9161_cos__paired,axiom,
    ! [Xa: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ot,Xa)),aa_real_real(cos_real,Xa)) ).

% cos_paired
tff(fact_9162_le__arcsin__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xa))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xa) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9163_arcsin__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xa)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9164_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_9165_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_9166_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K2)),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_9167_sin__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xa)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_9168_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_9169_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_9170_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B))) = aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_9171_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_9172_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_9173_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_9174_floor__eq,axiom,
    ! [Nb: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Nb ) ) ) ).

% floor_eq
tff(fact_9175_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_9176_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_9177_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_9178_floor__eq2,axiom,
    ! [Nb: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Nb ) ) ) ).

% floor_eq2
tff(fact_9179_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(divide_divide_int(aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_9180_floor__log__eq__powr__iff,axiom,
    ! [Xa: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(K),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_9181_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9182_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9183_fact__eq__fact__times,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cq),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb)),Ma))) ) ) ).

% fact_eq_fact_times
tff(fact_9184_fact__div__fact,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Ma)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cq),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Ma)) ) ) ).

% fact_div_fact
tff(fact_9185_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),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,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_9186_Maclaurin__sin__bound,axiom,
    ! [Xa: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xa)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_oo,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xa)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_9187_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T5)),aa_real_real(sin_real,T5)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9188_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_9189_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_9190_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_9191_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_9192_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_9193_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_9194_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_9195_complex__eq__cancel__iff2,axiom,
    ! [Xa: real,Y: real,Xaa: real] :
      ( ( aa_real_complex(complex2(Xa),Y) = aa_real_complex(real_V4546457046886955230omplex,Xaa) )
    <=> ( ( Xa = Xaa )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_9196_complex__of__real__code,axiom,
    ! [X: real] : aa_real_complex(real_V4546457046886955230omplex,X) = aa_real_complex(complex2(X),zero_zero_real) ).

% complex_of_real_code
tff(fact_9197_complex__of__real__def,axiom,
    ! [R2: real] : aa_real_complex(real_V4546457046886955230omplex,R2) = aa_real_complex(complex2(R2),zero_zero_real) ).

% complex_of_real_def
tff(fact_9198_Complex__mult__complex__of__real,axiom,
    ! [Xa: real,Y: real,R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(Xa),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),R2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_9199_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xa: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xa),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_9200_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_9201_Suc__times__binomial,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ).

% Suc_times_binomial
tff(fact_9202_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_9203_choose__mult__lemma,axiom,
    ! [Ma: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),R2)),Ma)) ).

% choose_mult_lemma
tff(fact_9204_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_9205_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = aa_num_real(numeral_numeral_real,W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_9206_divide__real__def,axiom,
    ! [Xa: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xa),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_9207_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
tff(fact_9208_zero__complex_Ocode,axiom,
    zero_zero_complex = aa_real_complex(complex2(zero_zero_real),zero_zero_real) ).

% zero_complex.code
tff(fact_9209_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ).

% zero_less_binomial
tff(fact_9210_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_9211_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_9212_choose__mult,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Ma),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K))) ) ) ) ).

% choose_mult
tff(fact_9213_binomial__absorb__comp,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_9214_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_9215_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ).

% inverse_powr
tff(fact_9216_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ).

% complex_mult
tff(fact_9217_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_9218_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_9219_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_9220_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D3: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D3),E)
         => ( aa_real_o(P,D3)
           => aa_real_o(P,E) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_9221_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_9222_forall__pos__mono,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D3: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D3),E)
         => ( aa_real_o(P,D3)
           => aa_real_o(P,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_9223_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))),E2) ) ) ).

% real_arch_inverse
tff(fact_9224_sqrt__divide__self__eq,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xa)),Xa) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9225_ln__inverse,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_inverse
tff(fact_9226_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_9227_prod__int__plus__eq,axiom,
    ! [Ia: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J))) = groups1705073143266064639nt_int(aTP_Lamm_cr,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,Ia),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),J)))) ).

% prod_int_plus_eq
tff(fact_9228_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_9229_binomial__mono,axiom,
    ! [K: nat,K5: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K5)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K5)) ) ) ).

% binomial_mono
tff(fact_9230_binomial__antimono,axiom,
    ! [K: nat,K5: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K5)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_9231_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_9232_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_9233_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_9234_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_9235_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_9236_log__inverse,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xa)) ) ) ) ) ).

% log_inverse
tff(fact_9237_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_9238_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K5)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_9239_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K5)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K5)) ) ) ).

% binomial_strict_mono
tff(fact_9240_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_9241_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_9242_complex__norm,axiom,
    ! [Xa: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xa),Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_9243_exp__plus__inverse__exp,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))) ).

% exp_plus_inverse_exp
tff(fact_9244_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_9245_plus__inverse__ge__2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(inverse_inverse_real,Xa))) ) ).

% plus_inverse_ge_2
tff(fact_9246_real__inv__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xa) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9247_tan__cot,axiom,
    ! [Xa: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xa)) ).

% tan_cot
tff(fact_9248_real__le__x__sinh,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9249_real__le__abs__sinh,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9250_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_9251_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_9252_divide__complex__def,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xa),Y) = aa_complex_complex(times_times_complex(Xa),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_9253_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A4: complex,R: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_complex_complex(exp_complex,A4)) ).

% complex_exp_exists
tff(fact_9254_atMost__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb) ).

% atMost_atLeast0
tff(fact_9255_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_9256_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_9257_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_9258_sum__choose__upper,axiom,
    ! [Ma: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ou,Ma)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Ma)) ).

% sum_choose_upper
tff(fact_9259_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ov,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_9260_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Ma: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ow,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9261_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Ma: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ow,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)),one_one_nat)),Ma) ).

% choose_rising_sum(2)
tff(fact_9262_sum__choose__diagonal,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ox(Ma),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Ma) ) ) ).

% sum_choose_diagonal
tff(fact_9263_vandermonde,axiom,
    ! [Ma: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oy(Ma),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),R2) ).

% vandermonde
tff(fact_9264_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% choose_row_sum
tff(fact_9265_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oz(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_9266_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9267_polynomial__product__nat,axiom,
    ! [Ma: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xa: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),I2)
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_pa(A),Xa)),aa_nat_set_nat(set_ord_atMost_nat,Ma))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pa(B),Xa)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_pc(A),B),Xa)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_9268_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pd,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb) ).

% choose_square_sum
tff(fact_9269_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9270_binomial__r__part__sum,axiom,
    ! [Ma: 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))),Ma)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_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(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ).

% binomial_r_part_sum
tff(fact_9271_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pe,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_9272_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_9273_real__scaleR__def,axiom,
    ! [A: real,Xa: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Xa) ).

% real_scaleR_def
tff(fact_9274_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] : aa_complex_complex(real_V2046097035970521341omplex(R2),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),B)) ).

% complex_scaleR
tff(fact_9275_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_9276_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_9277_sinh__real__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sinh_real,Xa) = zero_zero_real )
    <=> ( Xa = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_9278_sinh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_9279_sinh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% sinh_real_pos_iff
tff(fact_9280_sinh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_9281_sinh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% sinh_real_nonneg_iff
tff(fact_9282_complex__i__mult__minus,axiom,
    ! [Xa: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),Xa)) = aa_complex_complex(uminus1482373934393186551omplex,Xa) ).

% complex_i_mult_minus
tff(fact_9283_divide__i,axiom,
    ! [Xa: complex] : aa_complex_complex(divide1717551699836669952omplex(Xa),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),Xa) ).

% divide_i
tff(fact_9284_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_9285_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_9286_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9287_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_9288_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_9289_i__even__power,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_9290_cosh__real__nonzero,axiom,
    ! [Xa: real] : aa_real_real(cosh_real,Xa) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_9291_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
tff(fact_9292_cosh__real__pos,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_pos
tff(fact_9293_cosh__real__nonpos__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9294_cosh__real__nonneg__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9295_cosh__real__nonneg,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_nonneg
tff(fact_9296_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Z )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_9297_cosh__real__strict__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_9298_cosh__real__nonneg__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9299_cosh__real__nonpos__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9300_arcosh__cosh__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xa)) = Xa ) ) ).

% arcosh_cosh_real
tff(fact_9301_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9302_Complex__eq__i,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_complex(complex2(Xa),Y) = imaginary_unit )
    <=> ( ( Xa = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9303_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_9304_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_9305_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_9306_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(zero_zero_real),R2) ).

% i_complex_of_real
tff(fact_9307_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R2) ).

% complex_of_real_i
tff(fact_9308_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R: real,A4: real] : Z = 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,A4))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A4))))) ).

% complex_split_polar
tff(fact_9309_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_9310_cmod__complex__polar,axiom,
    ! [R2: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_9311_cosh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(inverse_inverse_real,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9312_sinh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),aa_real_real(inverse_inverse_real,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9313_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_9314_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9315_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9316_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9317_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% csqrt_eq_0
tff(fact_9318_csqrt__0,axiom,
    aa_complex_complex(csqrt,zero_zero_complex) = zero_zero_complex ).

% csqrt_0
tff(fact_9319_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9320_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9321_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9322_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_9323_cis__neq__zero,axiom,
    ! [A: real] : aa_real_complex(cis,A) != zero_zero_complex ).

% cis_neq_zero
tff(fact_9324_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).

% cis_mult
tff(fact_9325_Arg__zero,axiom,
    aa_complex_real(arg,zero_zero_complex) = zero_zero_real ).

% Arg_zero
tff(fact_9326_DeMoivre,axiom,
    ! [A: real,Nb: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre
tff(fact_9327_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_9328_of__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,Xa)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,Xa)) ) ) ).

% of_real_sqrt
tff(fact_9329_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_pf(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)),collect_complex(aTP_Lamm_ec(Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9330_cot__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xa)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_9331_cot__periodic,axiom,
    ! [Xa: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),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,Xa) ).

% cot_periodic
tff(fact_9332_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_pg,Y)) ).

% arctan_def
tff(fact_9333_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_9334_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9335_ln__neg__is__const,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xa) = the_real(aTP_Lamm_bf) ) ) ).

% ln_neg_is_const
tff(fact_9336_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_ph,Y)) ).

% arccos_def
tff(fact_9337_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_pi) ).

% pi_half
tff(fact_9338_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_pi)) ).

% pi_def
tff(fact_9339_cot__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xa)) ) ) ).

% cot_gt_zero
tff(fact_9340_tan__cot_H,axiom,
    ! [Xa: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)) = aa_real_real(cot_real,Xa) ).

% tan_cot'
tff(fact_9341_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_pj,Y)) ).

% arcsin_def
tff(fact_9342_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aTP_Lamm_ec(Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_pk(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9343_modulo__int__unfold,axiom,
    ! [K: int,Ma: nat,L: int,Nb: nat] :
      aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Ma))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Ma)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_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,Ma),Nb))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))))) ) ).

% modulo_int_unfold
tff(fact_9344_powr__int,axiom,
    ! [Xa: real,Ia: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_int_real(ring_1_of_int_real,Ia)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ia),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,Ia)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Ia))))) ) ) ).

% powr_int
tff(fact_9345_divide__int__unfold,axiom,
    ! [K: int,Ma: nat,L: int,Nb: nat] :
      aa_int_int(divide_divide_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Ma))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma)))))) ) ).

% divide_int_unfold
tff(fact_9346_real__root__zero,axiom,
    ! [Nb: nat] : aa_real_real(aa_nat_fun_real_real(root,Nb),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9347_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_9348_real__root__Suc__0,axiom,
    ! [Xa: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xa) = Xa ).

% real_root_Suc_0
tff(fact_9349_real__root__eq__iff,axiom,
    ! [Nb: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xa) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y) )
      <=> ( Xa = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9350_root__0,axiom,
    ! [Xa: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xa) = zero_zero_real ).

% root_0
tff(fact_9351_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9352_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xa) = zero_zero_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9353_real__root__less__iff,axiom,
    ! [Nb: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ).

% real_root_less_iff
tff(fact_9354_nat__0__iff,axiom,
    ! [Ia: int] :
      ( ( aa_int_nat(nat2,Ia) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),zero_zero_int) ) ).

% nat_0_iff
tff(fact_9355_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_9356_real__root__le__iff,axiom,
    ! [Nb: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% real_root_le_iff
tff(fact_9357_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_9358_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xa) = one_one_real )
      <=> ( Xa = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9359_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9360_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_9361_nat__zminus__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_9362_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_9363_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(times_times_int(K),aa_int_int(sgn_sgn_int,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9364_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,R2)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9365_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(L),aa_int_int(sgn_sgn_int,R2))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9366_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,R2)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9367_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9368_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9369_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9370_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9371_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_9372_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9373_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9374_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9375_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9376_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_9377_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_9378_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9379_nat__ceiling__le__eq,axiom,
    ! [Xa: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xa))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_9380_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_9381_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),Nb) = Xa ) ) ) ).

% real_root_pow_pos2
tff(fact_9382_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_9383_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9384_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9385_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9386_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9387_real__root__mult__exp,axiom,
    ! [Ma: nat,Nb: nat,Xa: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),Xa) = aa_real_real(aa_nat_fun_real_real(root,Ma),aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)) ).

% real_root_mult_exp
tff(fact_9388_real__root__mult,axiom,
    ! [Nb: nat,Xa: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).

% real_root_mult
tff(fact_9389_real__root__pos__pos__le,axiom,
    ! [Xa: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)) ) ).

% real_root_pos_pos_le
tff(fact_9390_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_9391_nat__numeral__as__int,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,X) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X)) ).

% nat_numeral_as_int
tff(fact_9392_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L2: int] : K != aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_9393_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z6) )
        <=> ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_9394_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_1: nat] : aa_nat_o(P,X_1)
    <=> ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% all_nat
tff(fact_9395_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
          & aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% ex_nat
tff(fact_9396_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_9397_unset__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),Nb) = aa_int_nat(nat2,aa_int_int(bit_se4203085406695923979it_int(Ma),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_9398_nat__mask__eq,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% nat_mask_eq
tff(fact_9399_real__root__less__mono,axiom,
    ! [Nb: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_9400_real__root__le__mono,axiom,
    ! [Nb: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_9401_real__root__power,axiom,
    ! [Nb: nat,Xa: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),K) ) ) ).

% real_root_power
tff(fact_9402_real__root__abs,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xa)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)) ) ) ).

% real_root_abs
tff(fact_9403_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_9404_zless__nat__eq__int__zless,axiom,
    ! [Ma: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_9405_int__eq__iff,axiom,
    ! [Ma: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ma) = Z )
    <=> ( ( Ma = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_9406_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_9407_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_9408_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(modulo_modulo_int(K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9409_int__minus,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Ma)))) ).

% int_minus
tff(fact_9410_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(times_times_int(W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_9411_and__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Ma),Nb) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_9412_nat__plus__as__int,axiom,
    ! [X: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Xa2) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_plus_as_int
tff(fact_9413_or__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Ma),Nb) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9414_nat__times__as__int,axiom,
    ! [X: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Xa2) = aa_int_nat(nat2,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_times_as_int
tff(fact_9415_real__nat__ceiling__ge,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xa)))) ).

% real_nat_ceiling_ge
tff(fact_9416_nat__minus__as__int,axiom,
    ! [X: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_minus_as_int
tff(fact_9417_nat__div__as__int,axiom,
    ! [X: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2) = aa_int_nat(nat2,aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_div_as_int
tff(fact_9418_nat__mod__as__int,axiom,
    ! [X: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2) = aa_int_nat(nat2,aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_mod_as_int
tff(fact_9419_real__root__gt__zero,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)) ) ) ).

% real_root_gt_zero
tff(fact_9420_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N3),Xa)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9421_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9422_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9423_zsgn__def,axiom,
    ! [Ia: int] :
      aa_int_int(sgn_sgn_int,Ia) = $ite(
        Ia = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ia),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_9424_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_9425_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_9426_nat__eq__iff2,axiom,
    ! [Ma: nat,W: int] :
      ( ( Ma = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,Ma),Ma = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_9427_nat__eq__iff,axiom,
    ! [W: int,Ma: nat] :
      ( ( aa_int_nat(nat2,W) = Ma )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,Ma),Ma = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_9428_split__nat,axiom,
    ! [P: fun_nat_o,Ia: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,Ia))
    <=> ( ! [N4: nat] :
            ( ( Ia = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => aa_nat_o(P,N4) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ia),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_9429_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ) ).

% le_nat_iff
tff(fact_9430_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
       => ( aa_int_nat(nat2,aa_int_int(plus_plus_int(Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_add_distrib
tff(fact_9431_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_9432_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ).

% nat_mult_distrib
tff(fact_9433_Suc__as__int,axiom,
    ! [X: nat] : aa_nat_nat(suc,X) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X)),one_one_int)) ).

% Suc_as_int
tff(fact_9434_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z6),Z)
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_diff_distrib
tff(fact_9435_nat__diff__distrib_H,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9436_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_9437_nat__div__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_nat(nat2,aa_int_int(divide_divide_int(Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_9438_nat__div__distrib_H,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(divide_divide_int(Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_9439_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_9440_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_9441_nat__floor__neg,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9442_nat__mod__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(modulo_modulo_int(Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_9443_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_9444_floor__eq3,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = Nb ) ) ) ).

% floor_eq3
tff(fact_9445_le__nat__floor,axiom,
    ! [Xa: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_9446_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_9447_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9448_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9449_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Nb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ) ).

% bit_nat_iff
tff(fact_9450_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(divide_divide_int(K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_9451_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(modulo_modulo_int(K),L) = $ite(
        L = zero_zero_int,
        K,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_9452_real__root__pos__pos,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)) ) ) ).

% real_root_pos_pos
tff(fact_9453_real__root__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_real_real(aa_nat_fun_real_real(root,N3),Xa)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9454_real__root__decreasing,axiom,
    ! [Nb: nat,N3: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N3),Xa)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)) ) ) ) ).

% real_root_decreasing
tff(fact_9455_real__root__pow__pos,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),Nb) = Xa ) ) ) ).

% real_root_pow_pos
tff(fact_9456_real__root__power__cancel,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)) = Xa ) ) ) ).

% real_root_power_cancel
tff(fact_9457_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xa )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xa) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9458_odd__real__root__pow,axiom,
    ! [Nb: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),Nb) = Xa ) ) ).

% odd_real_root_pow
tff(fact_9459_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xa )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xa) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9460_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb)) = Xa ) ) ).

% odd_real_root_power_cancel
tff(fact_9461_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_9462_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_9463_nat__less__iff,axiom,
    ! [W: int,Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),Ma)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,Ma)) ) ) ).

% nat_less_iff
tff(fact_9464_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9465_floor__eq4,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = Nb ) ) ) ).

% floor_eq4
tff(fact_9466_diff__nat__eq__if,axiom,
    ! [Z: int,Z6: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z6),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(minus_minus_int(Z),Z6),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_9467_real__root__increasing,axiom,
    ! [Nb: nat,N3: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_real_real(aa_nat_fun_real_real(root,N3),Xa)) ) ) ) ) ).

% real_root_increasing
tff(fact_9468_nat__dvd__iff,axiom,
    ! [Z: int,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),Ma)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,Ma)),Ma = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_9469_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q2),L)),R2) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q2,R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9470_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9471_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9472_log__base__root,axiom,
    ! [Nb: nat,B: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xa)) ) ) ) ).

% log_base_root
tff(fact_9473_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L3: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K3 = aa_int_int(times_times_int(Q4),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = product_Pair_int_int(Q4,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L3))
            & ( K3 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q4),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9474_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = product_Pair_int_int(Q3,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(times_times_int(Q3),A22) ) ) )
         => ~ ! [R: int,Q3: int] :
                ( ( A32 = product_Pair_int_int(Q3,R) )
               => ( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q3),A22)),R) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9475_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9476_root__powr__inverse,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xa) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9477_even__nat__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ) ).

% even_nat_iff
tff(fact_9478_powr__real__of__int,axiom,
    ! [Xa: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_9479_arctan__inverse,axiom,
    ! [Xa: real] :
      ( ( Xa != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xa)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xa)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xa)) ) ) ).

% arctan_inverse
tff(fact_9480_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9481_sgn__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_9482_zero__le__sgn__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% zero_le_sgn_iff
tff(fact_9483_sgn__root,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)) = aa_real_real(sgn_sgn_real,Xa) ) ) ).

% sgn_root
tff(fact_9484_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_complex(cis,aa_complex_real(arg,Z)) = aa_complex_complex(sgn_sgn_complex,Z) ) ) ).

% cis_Arg
tff(fact_9485_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9486_sin__times__pi__eq__0,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),pi)) = zero_zero_real )
    <=> aa_set_real_o(member_real(Xa),ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9487_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xa: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = Xa )
     => ( ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9488_sgn__power__root,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa))),Nb)) = Xa ) ) ).

% sgn_power_root
tff(fact_9489_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_9490_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xa: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),Nb)) = Xa )
             => aa_real_o(P,Y5) ) ) ) ) ).

% split_root
tff(fact_9491_floor__real__def,axiom,
    ! [Xa: real] : aa_real_int(archim6058952711729229775r_real,Xa) = the_int(aTP_Lamm_pl(Xa)) ).

% floor_real_def
tff(fact_9492_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9493_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9494_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9495_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_pm(Z))) ).

% Arg_def
tff(fact_9496_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_9497_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_9498_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_9499_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_9500_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ).

% xor_nat_numerals(3)
tff(fact_9501_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa)) ).

% xor_nat_numerals(4)
tff(fact_9502_xor__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Ma),Nb) = $ite(
        Ma = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Ma,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,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9503_xor__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9504_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_9505_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9506_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9507_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa_int_int(bit_concat_bit(Nb,zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(Nb),L) ).

% concat_bit_of_zero_1
tff(fact_9508_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9509_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_9510_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9511_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2159334234014336723it_int(Nb),K) = aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% flip_bit_int_def
tff(fact_9512_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se6526347334894502574or_int(K),L)),Nb)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        <=> aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_9513_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(Nb),K)) ).

% push_bit_nat_eq
tff(fact_9514_XOR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(Xa),Y)) ) ) ).

% XOR_lower
tff(fact_9515_set__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Ma),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9516_flip__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Ma),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9517_bit__push__bit__iff__int,axiom,
    ! [Ma: nat,K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(Ma),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9518_xor__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Ma),Nb) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_9519_bit__push__bit__iff__nat,axiom,
    ! [Ma: nat,Q2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Ma),Q2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9520_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(plus_plus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_eq
tff(fact_9521_concat__bit__def,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_def
tff(fact_9522_set__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(Nb),K) = aa_int_int(bit_se1409905431419307370or_int(K),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% set_bit_int_def
tff(fact_9523_push__bit__nat__def,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9524_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Nb),K) = aa_int_int(times_times_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_int_def
tff(fact_9525_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_minus_one
tff(fact_9526_XOR__upper,axiom,
    ! [Xa: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(Xa),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9527_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9528_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K),L) = $ite(
        K = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9529_Sum__Ico__nat,axiom,
    ! [Ma: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cq),aa_nat_set_nat(set_or4665077453230672383an_nat(Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9530_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_9531_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_9532_atLeastLessThan__singleton,axiom,
    ! [Ma: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Ma),aa_nat_nat(suc,Ma)) = insert_nat(Ma,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9533_or__minus__minus__numerals,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Ma)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9534_and__minus__minus__numerals,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se1409905431419307370or_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Ma)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9535_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% bit_not_int_iff
tff(fact_9536_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
          & aa_nat_o(P,M3) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less_eq
tff(fact_9537_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
         => aa_nat_o(P,M3) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less_eq
tff(fact_9538_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_9539_lessThan__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ).

% lessThan_atLeast0
tff(fact_9540_atLeastLessThan0,axiom,
    ! [Ma: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Ma),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9541_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9542_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9543_and__not__numerals_I1_J,axiom,
    aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9544_or__not__numerals_I1_J,axiom,
    aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9545_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_9546_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(Nb),K) = aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int))) ).

% unset_bit_int_def
tff(fact_9547_xor__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).

% xor_int_def
tff(fact_9548_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9549_atLeastLessThan__add__Un,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(Ia),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(Ia),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9550_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(divide_divide_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9551_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_not_iff_int
tff(fact_9552_and__not__numerals_I4_J,axiom,
    ! [Ma: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma)) ).

% and_not_numerals(4)
tff(fact_9553_and__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9554_or__not__numerals_I4_J,axiom,
    ! [Ma: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9555_or__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_9556_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K)),Nb)
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(minus_minus_int(K),one_one_int))),Nb) ) ).

% bit_minus_int_iff
tff(fact_9557_int__numeral__or__not__num__neg,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Ma),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9558_int__numeral__not__or__num__neg,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Ma))) ).

% int_numeral_not_or_num_neg
tff(fact_9559_numeral__or__not__num__eq,axiom,
    ! [Ma: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Ma),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9560_atLeastLessThanSuc,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Ma),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb),insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(Ma),Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9561_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9562_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9563_and__not__numerals_I5_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9564_and__not__numerals_I7_J,axiom,
    ! [Ma: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma)) ).

% and_not_numerals(7)
tff(fact_9565_or__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_9566_and__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9567_or__not__numerals_I7_J,axiom,
    ! [Ma: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9568_atLeastLessThan__nat__numeral,axiom,
    ! [Ma: nat,K: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Ma),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(Ma),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9569_and__not__numerals_I9_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9570_and__not__numerals_I6_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9571_or__not__numerals_I6_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9572_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9573_or__not__numerals_I5_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9574_and__not__numerals_I8_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9575_or__not__numerals_I9_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9576_or__not__numerals_I8_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9577_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9578_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9579_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J3)) ) )
     => ( ! [I2: nat,J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J3)),aa_nat_nat(B,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pn(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9580_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa_set_int_o(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_9581_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J2: nat] :
        ? [M7: nat] :
        ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M3)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X6,M3)),aa_nat_real(X6,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J2)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9582_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_9583_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_9584_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9585_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9586_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( aa_complex_real(re,Y) = zero_zero_real )
       => ( aa_real_real(cos_real,aa_complex_real(arg,Y)) = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_9587_imaginary__unit_Osimps_I1_J,axiom,
    aa_complex_real(re,imaginary_unit) = zero_zero_real ).

% imaginary_unit.simps(1)
tff(fact_9588_zero__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(1)
tff(fact_9589_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xa: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R2),Xa)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xa)) ).

% scaleR_complex.simps(1)
tff(fact_9590_Re__csqrt,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))) ).

% Re_csqrt
tff(fact_9591_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real)
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9592_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_9593_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
            $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9594_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9595_csqrt__of__real__nonpos,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_real(im,Xa) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xa)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xa) = 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,Xa))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9596_complex__Im__fact,axiom,
    ! [Nb: nat] : aa_complex_real(im,aa_nat_complex(semiri5044797733671781792omplex,Nb)) = zero_zero_real ).

% complex_Im_fact
tff(fact_9597_complex__Im__of__int,axiom,
    ! [Z: int] : aa_complex_real(im,aa_int_complex(ring_17405671764205052669omplex,Z)) = zero_zero_real ).

% complex_Im_of_int
tff(fact_9598_complex__Im__of__nat,axiom,
    ! [Nb: nat] : aa_complex_real(im,aa_nat_complex(semiri8010041392384452111omplex,Nb)) = zero_zero_real ).

% complex_Im_of_nat
tff(fact_9599_Im__complex__of__real,axiom,
    ! [Z: real] : aa_complex_real(im,aa_real_complex(real_V4546457046886955230omplex,Z)) = zero_zero_real ).

% Im_complex_of_real
tff(fact_9600_Im__power__real,axiom,
    ! [Xa: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xa) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(power_power_complex(Xa),Nb)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_9601_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_9602_Im__i__times,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_complex_real(re,Z) ).

% Im_i_times
tff(fact_9603_Re__power__real,axiom,
    ! [Xa: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xa) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(Xa),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xa)),Nb) ) ) ).

% Re_power_real
tff(fact_9604_Re__i__times,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).

% Re_i_times
tff(fact_9605_csqrt__of__real__nonneg,axiom,
    ! [Xa: complex] :
      ( ( aa_complex_real(im,Xa) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xa))
       => ( aa_complex_complex(csqrt,Xa) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,Xa))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_9606_csqrt__minus,axiom,
    ! [Xa: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xa)),zero_zero_real)
        | ( ( aa_complex_real(im,Xa) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xa)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xa)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xa)) ) ) ).

% csqrt_minus
tff(fact_9607_zero__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(2)
tff(fact_9608_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9609_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xa: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R2),Xa)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xa)) ).

% scaleR_complex.simps(2)
tff(fact_9610_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),ring_1_Ints_complex)
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I: int] : aa_complex_real(re,Z) = aa_int_real(ring_1_of_int_real,I) ) ) ).

% complex_is_Int_iff
tff(fact_9611_times__complex_Osimps_I2_J,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xa)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xa)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_9612_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(im,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(re,Z)) ) ) ).

% cmod_eq_Re
tff(fact_9613_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(re,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(im,Z)) ) ) ).

% cmod_eq_Im
tff(fact_9614_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_real_real(abs_abs_real,aa_complex_real(re,Z)) = aa_complex_real(real_V1022390504157884413omplex,Z) )
     => ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% Im_eq_0
tff(fact_9615_times__complex_Osimps_I1_J,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xa)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xa)),aa_complex_real(im,Y))) ).

% times_complex.simps(1)
tff(fact_9616_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xa: complex] : aa_complex_complex(real_V2046097035970521341omplex(R2),Xa) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xa))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xa))) ).

% scaleR_complex.code
tff(fact_9617_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z))) ) ) ).

% csqrt_principal
tff(fact_9618_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_9619_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_9620_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_9621_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_9622_times__complex_Ocode,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_complex(times_times_complex(Xa),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xa)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xa)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xa)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xa)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_9623_exp__eq__polar,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,Z) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z)))),aa_real_complex(cis,aa_complex_real(im,Z))) ).

% exp_eq_polar
tff(fact_9624_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9625_Im__power2,axiom,
    ! [Xa: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(Xa),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,Xa))),aa_complex_real(im,Xa)) ).

% Im_power2
tff(fact_9626_Re__power2,axiom,
    ! [Xa: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9627_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9628_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9629_inverse__complex_Osimps_I1_J,axiom,
    ! [Xa: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9630_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_9631_Re__divide,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xa)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xa)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9632_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_9633_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9634_inverse__complex_Osimps_I2_J,axiom,
    ! [Xa: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9635_Im__divide,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xa)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xa)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9636_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_9637_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9638_inverse__complex_Ocode,axiom,
    ! [Xa: complex] : aa_complex_complex(invers8013647133539491842omplex,Xa) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xa)),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,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9639_Complex__divide,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xa),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xa)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xa)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xa)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xa)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9640_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9641_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R2)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9642_real__eq__imaginary__iff,axiom,
    ! [Y: complex,Xa: complex] :
      ( aa_set_complex_o(member_complex(Y),real_V2521375963428798218omplex)
     => ( aa_set_complex_o(member_complex(Xa),real_V2521375963428798218omplex)
       => ( ( Xa = aa_complex_complex(times_times_complex(imaginary_unit),Y) )
        <=> ( ( Xa = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9643_imaginary__eq__real__iff,axiom,
    ! [Y: complex,Xa: complex] :
      ( aa_set_complex_o(member_complex(Y),real_V2521375963428798218omplex)
     => ( aa_set_complex_o(member_complex(Xa),real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y) = Xa )
        <=> ( ( Xa = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9644_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),real_V2521375963428798218omplex)
    <=> ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% complex_is_Real_iff
tff(fact_9645_Complex__in__Reals,axiom,
    ! [Xa: real] : aa_set_complex_o(member_complex(aa_real_complex(complex2(Xa),zero_zero_real)),real_V2521375963428798218omplex) ).

% Complex_in_Reals
tff(fact_9646_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9647_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_gr(L),Qr) ).

% divmod_step_integer_def
tff(fact_9648_complex__cnj__mult,axiom,
    ! [Xa: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(times_times_complex(Xa),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Xa)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_mult
tff(fact_9649_complex__cnj__zero,axiom,
    aa_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).

% complex_cnj_zero
tff(fact_9650_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
tff(fact_9651_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_9652_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = $ite(
        K = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9653_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9654_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_9655_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_9656_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_9657_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9658_divmod__integer_H__def,axiom,
    ! [Ma: num,Nb: num] : unique3479559517661332726nteger(Ma,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb))) ).

% divmod_integer'_def
tff(fact_9659_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9660_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9661_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9662_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_9663_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_9664_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_9665_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_9666_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_gt_0
tff(fact_9667_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_9668_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_9669_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_ge_0
tff(fact_9670_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_gt_0
tff(fact_9671_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_9672_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_9673_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_ge_0
tff(fact_9674_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9675_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ) ).

% complex_div_gt_0
tff(fact_9676_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_9677_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9678_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9679_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_9680_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9681_integer__of__int__code,axiom,
    ! [K: int] :
      aa_int_Code_integer(code_integer_of_int,K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9682_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ea,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_9683_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_9684_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_po,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_9685_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_9686_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_9687_abs__integer__code,axiom,
    ! [K: code_integer] :
      abs_abs_Code_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ).

% abs_integer_code
tff(fact_9688_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : modulo364778990260209775nteger(aa_int_Code_integer(code_integer_of_int,Xa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(modulo_modulo_int(Xa),Xb)) ).

% modulo_integer.abs_eq
tff(fact_9689_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_9690_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_9691_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_pp),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9692_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_bh),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9693_times__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(times_times_int(Xa),Xb)) ).

% times_integer.abs_eq
tff(fact_9694_card__less__Suc2,axiom,
    ! [M6: set_nat,Ia: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_pq(M6),Ia))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_pr(M6),Ia))) ) ) ).

% card_less_Suc2
tff(fact_9695_card__less__Suc,axiom,
    ! [M6: set_nat,Ia: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_pq(M6),Ia)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_pr(M6),Ia))) ) ) ).

% card_less_Suc
tff(fact_9696_card__less,axiom,
    ! [M6: set_nat,Ia: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_pr(M6),Ia))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9697_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_9698_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))))
     => ( A2 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9699_less__eq__nat_Osimps_I2_J,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Ma)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9700_max__Suc2,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_ps,Nb)),Ma) ).

% max_Suc2
tff(fact_9701_max__Suc1,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Ma) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_pt,Nb)),Ma) ).

% max_Suc1
tff(fact_9702_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N3)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9703_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cq),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cq),S2)) ).

% card_sum_le_nat_sum
tff(fact_9704_card__nth__roots,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_pk(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_9705_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aTP_Lamm_ec(Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_9706_diff__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cq),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) ).

% diff_Suc
tff(fact_9707_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K,numera6620942414471956472nteger(aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K)) ).

% bit_cut_integer_def
tff(fact_9708_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_pu(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9709_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K,L),modulo364778990260209775nteger(K,L)) ).

% divmod_integer_def
tff(fact_9710_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_9711_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,abs_abs_Code_integer(J)) ).

% divmod_abs_code(5)
tff(fact_9712_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)),modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))) ).

% divmod_abs_def
tff(fact_9713_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_pv(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_pw(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9714_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cq),Nat) ).

% pred_def
tff(fact_9715_bezw__0,axiom,
    ! [Xa: nat] : bezw(Xa,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_9716_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9717_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_9718_prod__decode__aux_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xa)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9719_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9720_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9721_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9722_snd__divmod__nat,axiom,
    ! [Ma: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(Ma,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) ).

% snd_divmod_nat
tff(fact_9723_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9724_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9725_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9726_drop__bit__push__bit__int,axiom,
    ! [Ma: nat,Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Ma),aa_int_int(bit_se545348938243370406it_int(Nb),K)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)),K)) ).

% drop_bit_push_bit_int
tff(fact_9727_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),K) = aa_int_int(divide_divide_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_int_def
tff(fact_9728_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Ma: nat] :
      nat_prod_decode_aux(K,Ma) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Ma)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,K)))) ).

% prod_decode_aux.simps
tff(fact_9729_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_9730_vebt__maxt_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      (B4),
                      some_nat(one_one_nat),
                      $ite((A4),some_nat(zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Ma2) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9731_vebt__mint_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      (A4),
                      some_nat(zero_zero_nat),
                      $ite((B4),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Mi2) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9732_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_integer(K,L)) = modulo364778990260209775nteger(K,L) ).

% snd_divmod_integer
tff(fact_9733_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_abs(K,L)) = modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)) ).

% snd_divmod_abs
tff(fact_9734_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9735_fst__divmod__nat,axiom,
    ! [Ma: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(Ma,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) ).

% fst_divmod_nat
tff(fact_9736_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(Nb),K)) ).

% drop_bit_nat_eq
tff(fact_9737_drop__bit__nat__def,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9738_minus__one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_mod_numeral
tff(fact_9739_one__mod__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(one,Nb)))) ).

% one_mod_minus_numeral
tff(fact_9740_numeral__mod__minus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(Ma,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_9741_minus__numeral__mod__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(Ma,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_9742_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      aa_int_int(adjust_mod(L),R2) = $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(L),R2)) ).

% Divides.adjust_mod_def
tff(fact_9743_bezw__non__0,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(Xa,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y))),aa_int_int(minus_minus_int(product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)))),aa_int_int(times_times_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y))))) ) ) ).

% bezw_non_0
tff(fact_9744_bezw_Osimps,axiom,
    ! [Xa: nat,Y: nat] :
      bezw(Xa,Y) = $ite(Y = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y))),aa_int_int(minus_minus_int(product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)))),aa_int_int(times_times_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y)))))) ).

% bezw.simps
tff(fact_9745_bezw_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xa,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))),aa_int_int(minus_minus_int(product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(times_times_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_9746_bezw_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xa,Xaa) = Y )
     => ( accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))),aa_int_int(minus_minus_int(product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(times_times_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Xaa)))))) )
           => ~ accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_9747_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xa = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9748_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ! [Uv2: $o] :
              ( ( Xa = vEBT_Leaf($true,(Uv2)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xa = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9749_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9750_prod__decode__aux_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xa)))) )
           => ~ accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9751_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_px(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9752_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Ma: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
         => aa_nat_o(P,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
           => ( ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I3)
                 => aa_nat_o(P,I3) )
             => aa_nat_o(P,K2) ) )
       => aa_nat_o(P,Ma) ) ) ).

% nat_descend_induct
tff(fact_9753_xor__minus__numerals_I2_J,axiom,
    ! [K: int,Nb: num] : aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(K),neg_numeral_sub_int(Nb,one))) ).

% xor_minus_numerals(2)
tff(fact_9754_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(neg_numeral_sub_int(Nb,one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9755_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_9756_sub__BitM__One__eq,axiom,
    ! [Nb: num] : neg_numeral_sub_int(aa_num_num(bitM,Nb),one) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),neg_numeral_sub_int(Nb,one)) ).

% sub_BitM_One_eq
tff(fact_9757_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M3: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N4)
          & aa_set_nat_o(member_nat(N4),S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_9758_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M2)
         => ? [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N5)
              & aa_set_nat_o(member_nat(N5),S2) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S2) ) ).

% unbounded_k_infinite
tff(fact_9759_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [R: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S2)))
          & ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_set_nat_nat(finite_card_nat,S2))
             => aa_set_nat_o(member_nat(aa_nat_nat(R,N5)),S2) ) ) ) ).

% finite_enumerate
tff(fact_9760_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9761_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9762_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,code_nat_of_integer(K)) = ord_max_Code_integer(zero_z3403309356797280102nteger,K) ).

% of_nat_of_integer
tff(fact_9763_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9764_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(numera6620942414471956472nteger(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_9765_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_py,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9766_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_po,aTP_Lamm_ea) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9767_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_pz,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9768_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9769_times__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xaa)) = aa_int_int(times_times_int(code_int_of_integer(Xa)),code_int_of_integer(Xaa)) ).

% times_integer.rep_eq
tff(fact_9770_modulo__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : code_int_of_integer(modulo364778990260209775nteger(Xa,Xaa)) = aa_int_int(modulo_modulo_int(code_int_of_integer(Xa)),code_int_of_integer(Xaa)) ).

% modulo_integer.rep_eq
tff(fact_9771_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(times_times_int(abs_Integ(Xa)),abs_Integ(Xb)) = abs_Integ(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qb),Xa),Xb)) ).

% times_int.abs_eq
tff(fact_9772_zero__int__def,axiom,
    zero_zero_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9773_int__def,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Nb),zero_zero_nat)) ).

% int_def
tff(fact_9774_one__int__def,axiom,
    one_one_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9775_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,abs_Integ(Xa)),abs_Integ(Xb))
    <=> aa_Pro6150637604445964947_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_qd,Xa),Xb) ) ).

% less_int.abs_eq
tff(fact_9776_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,abs_Integ(Xa)),abs_Integ(Xb))
    <=> aa_Pro6150637604445964947_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_qf,Xa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_9777_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(plus_plus_int(abs_Integ(Xa)),abs_Integ(Xb)) = abs_Integ(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qh),Xa),Xb)) ).

% plus_int.abs_eq
tff(fact_9778_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(minus_minus_int(abs_Integ(Xa)),abs_Integ(Xb)) = abs_Integ(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qj),Xa),Xb)) ).

% minus_int.abs_eq
tff(fact_9779_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),aa_num_num(inc,aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_9780_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_9781_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9782_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_9783_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( aa_nat_num(num_of_nat,Nb) = one ) ) ).

% num_of_nat_One
tff(fact_9784_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_9785_num__of__nat__plus__distrib,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,Ma)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9786_less__eq__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_qf,rep_Integ(Xa)),rep_Integ(Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_9787_less__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_qd,rep_Integ(Xa)),rep_Integ(Xaa)) ) ).

% less_int.rep_eq
tff(fact_9788_pow_Osimps_I3_J,axiom,
    ! [Xa: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xa),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,Xa),Y))),Xa) ).

% pow.simps(3)
tff(fact_9789_Gcd__remove0__nat,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M6,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9790_sqr_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_num(sqr,aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,Nb))) ).

% sqr.simps(2)
tff(fact_9791_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9792_sqr__conv__mult,axiom,
    ! [Xa: num] : aa_num_num(sqr,Xa) = aa_num_num(aa_num_fun_num_num(times_times_num,Xa),Xa) ).

% sqr_conv_mult
tff(fact_9793_pow_Osimps_I2_J,axiom,
    ! [Xa: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xa),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,Xa),Y)) ).

% pow.simps(2)
tff(fact_9794_sqr_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_num(sqr,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(sqr,Nb)),Nb))) ).

% sqr.simps(3)
tff(fact_9795_Gcd__int__greater__eq__0,axiom,
    ! [K6: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K6)) ).

% Gcd_int_greater_eq_0
tff(fact_9796_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(aa_num_num(bit0,one)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9797_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xa: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qk,C),aa_nat_set_nat(set_or4665077453230672383an_nat(Xa),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9798_bij__betw__Suc,axiom,
    ! [M6: set_nat,N3: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N3)
    <=> ( image_nat_nat(suc,M6) = N3 ) ) ).

% bij_betw_Suc
tff(fact_9799_image__Suc__atLeastAtMost,axiom,
    ! [Ia: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(Ia),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Ia)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9800_image__Suc__atLeastLessThan,axiom,
    ! [Ia: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(Ia),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,Ia)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9801_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_9802_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb) ).

% image_Suc_lessThan
tff(fact_9803_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9804_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9805_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9806_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9807_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9808_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9809_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit0,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_9810_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_ql(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(minus_minus_int(U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9811_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9812_take__bit__numeral__minus__numeral__int,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_qm(Ma),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Ma),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9813_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = produc4245557441103728435nt_int(aTP_Lamm_qn,quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9814_take__bit__num__simps_I1_J,axiom,
    ! [Ma: num] : bit_take_bit_num(zero_zero_nat,Ma) = none_num ).

% take_bit_num_simps(1)
tff(fact_9815_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9816_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9817_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% rat_zero_code
tff(fact_9818_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa_rat_rat(divide_divide_rat(Q2),R2) = aa_rat_rat(times_times_rat(Q2),aa_rat_rat(inverse_inverse_rat,R2)) ).

% divide_rat_def
tff(fact_9819_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_qo,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9820_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P2,Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% quotient_of_denom_pos
tff(fact_9821_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_9822_take__bit__num__def,axiom,
    ! [Nb: nat,Ma: num] :
      bit_take_bit_num(Nb,Ma) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Ma)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Ma))))) ).

% take_bit_num_def
tff(fact_9823_and__minus__numerals_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(3)
tff(fact_9824_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Ma: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Ma)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(7)
tff(fact_9825_and__minus__numerals_I4_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_9826_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Ma)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Ma))) ).

% take_bit_num_simps(4)
tff(fact_9827_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Ma)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_qp,bit_take_bit_num(Nb,Ma)) ).

% take_bit_num_simps(3)
tff(fact_9828_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,Ma: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,Ma)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Ma))) ).

% take_bit_num_simps(7)
tff(fact_9829_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Ma: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Ma)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_qp,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Ma)) ).

% take_bit_num_simps(6)
tff(fact_9830_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Ma: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Ma)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_9831_sgn__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(sgn_sgn_rat,A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9832_abs__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_rat_def
tff(fact_9833_and__not__num_Osimps_I8_J,axiom,
    ! [Ma: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Ma),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qq,bit_and_not_num(Ma,Nb)) ).

% and_not_num.simps(8)
tff(fact_9834_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9835_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S)
           => ! [T5: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T5)
               => ( R2 != aa_rat_rat(plus_plus_rat(S),T5) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9836_and__not__num_Osimps_I4_J,axiom,
    ! [Ma: num] : bit_and_not_num(aa_num_num(bit0,Ma),one) = some_num(aa_num_num(bit0,Ma)) ).

% and_not_num.simps(4)
tff(fact_9837_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit0,Nb)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9838_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_9839_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Ma)) = case_nat_option_num(none_num,aTP_Lamm_qr(Ma),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9840_and__not__num_Osimps_I7_J,axiom,
    ! [Ma: num] : bit_and_not_num(aa_num_num(bit1,Ma),one) = some_num(aa_num_num(bit0,Ma)) ).

% and_not_num.simps(7)
tff(fact_9841_and__not__num__eq__Some__iff,axiom,
    ! [Ma: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(Ma,Nb) = some_num(Q2) )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9842_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Ma)) = case_nat_option_num(none_num,aTP_Lamm_qs(Ma),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9843_and__not__num__eq__None__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( bit_and_not_num(Ma,Nb) = none_num )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9844_int__numeral__not__and__num,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Ma)) ).

% int_numeral_not_and_num
tff(fact_9845_int__numeral__and__not__num,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,Nb)) ).

% int_numeral_and_not_num
tff(fact_9846_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_qu(Q2),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9847_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_qw(Q2),quotient_of(P2)) ).

% rat_times_code
tff(fact_9848_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_qy(Q2),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9849_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(product_Pair_int_int(P2,zero_zero_int)) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% normalize_denom_zero
tff(fact_9850_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q2),zero_zero_int)
     => ( normalize(product_Pair_int_int(P2,Q2)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P2),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_9851_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q2: int] :
      ( ( normalize(R2) = product_Pair_int_int(P2,Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% normalize_denom_pos
tff(fact_9852_normalize__crossproduct,axiom,
    ! [Q2: int,Sb: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( Sb != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P2,Q2)) = normalize(product_Pair_int_int(R2,Sb)) )
         => ( aa_int_int(times_times_int(P2),Sb) = aa_int_int(times_times_int(R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9853_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_ra(Q2),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9854_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      normalize(P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P2)),
        $let(
          a: int,
          a:= aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2)),
          product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P2)),a),aa_int_int(divide_divide_int(product_snd_int_int(P2)),a)) ),
        $ite(
          product_snd_int_int(P2) = zero_zero_int,
          product_Pair_int_int(zero_zero_int,one_one_int),
          $let(
            a: int,
            a:= aa_int_int(uminus_uminus_int,aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2))),
            product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P2)),a),aa_int_int(divide_divide_int(product_snd_int_int(P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9855_and__not__num_Oelims,axiom,
    ! [Xa: num,Xaa: num,Y: option_num] :
      ( ( bit_and_not_num(Xa,Xaa) = Y )
     => ( ( ( Xa = one )
         => ( ( Xaa = one )
           => ( Y != none_num ) ) )
       => ( ( ( Xa = one )
           => ( ? [N: num] : Xaa = aa_num_num(bit0,N)
             => ( Y != some_num(one) ) ) )
         => ( ( ( Xa = one )
             => ( ? [N: num] : Xaa = aa_num_num(bit1,N)
               => ( Y != none_num ) ) )
           => ( ! [M2: num] :
                  ( ( Xa = aa_num_num(bit0,M2) )
                 => ( ( Xaa = one )
                   => ( Y != some_num(aa_num_num(bit0,M2)) ) ) )
             => ( ! [M2: num] :
                    ( ( Xa = aa_num_num(bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xa = aa_num_num(bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Y != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xa = aa_num_num(bit1,M2) )
                       => ( ( Xaa = one )
                         => ( Y != some_num(aa_num_num(bit0,M2)) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xa = aa_num_num(bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qq,bit_and_not_num(M2,N)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xa = aa_num_num(bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Y != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9856_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(Nb,Ma) = produc478579273971653890on_num(aTP_Lamm_re,product_Pair_nat_num(Nb,Ma)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9857_gcd__pos__int,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(gcd_gcd_int(Ma),Nb))
    <=> ( ( Ma != zero_zero_int )
        | ( Nb != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9858_gcd__0__left__int,axiom,
    ! [Xa: int] : aa_int_int(gcd_gcd_int(zero_zero_int),Xa) = aa_int_int(abs_abs_int,Xa) ).

% gcd_0_left_int
tff(fact_9859_gcd__0__int,axiom,
    ! [Xa: int] : aa_int_int(gcd_gcd_int(Xa),zero_zero_int) = aa_int_int(abs_abs_int,Xa) ).

% gcd_0_int
tff(fact_9860_gcd__mult__distrib__int,axiom,
    ! [K: int,Ma: int,Nb: int] : aa_int_int(times_times_int(aa_int_int(abs_abs_int,K)),aa_int_int(gcd_gcd_int(Ma),Nb)) = aa_int_int(gcd_gcd_int(aa_int_int(times_times_int(K),Ma)),aa_int_int(times_times_int(K),Nb)) ).

% gcd_mult_distrib_int
tff(fact_9861_bezout__int,axiom,
    ! [Xa: int,Y: int] :
    ? [U2: int,V2: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(U2),Xa)),aa_int_int(times_times_int(V2),Y)) = aa_int_int(gcd_gcd_int(Xa),Y) ).

% bezout_int
tff(fact_9862_gcd__red__int,axiom,
    ! [Xa: int,Y: int] : aa_int_int(gcd_gcd_int(Xa),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(modulo_modulo_int(Xa),Y)) ).

% gcd_red_int
tff(fact_9863_gcd__ge__0__int,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(gcd_gcd_int(Xa),Y)) ).

% gcd_ge_0_int
tff(fact_9864_and__not__num_Osimps_I5_J,axiom,
    ! [Ma: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Ma),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_and_not_num(Ma,Nb)) ).

% and_not_num.simps(5)
tff(fact_9865_and__not__num_Osimps_I6_J,axiom,
    ! [Ma: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Ma),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Ma,Nb)) ).

% and_not_num.simps(6)
tff(fact_9866_and__not__num_Osimps_I9_J,axiom,
    ! [Ma: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Ma),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Ma,Nb)) ).

% and_not_num.simps(9)
tff(fact_9867_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B)),B) ) ).

% gcd_le2_int
tff(fact_9868_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B)),A) ) ).

% gcd_le1_int
tff(fact_9869_gcd__cases__int,axiom,
    ! [Xa: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,aa_int_int(gcd_gcd_int(Xa),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(gcd_gcd_int(Xa),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,Xa)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P,aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,Xa)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,aa_int_int(gcd_gcd_int(Xa),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9870_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(gcd_gcd_int(A),B) ) ) ).

% gcd_unique_int
tff(fact_9871_gcd__non__0__int,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
     => ( aa_int_int(gcd_gcd_int(Xa),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(modulo_modulo_int(Xa),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9872_gcd__code__int,axiom,
    ! [K: int,L: int] :
      aa_int_int(gcd_gcd_int(K),L) = aa_int_int(abs_abs_int,
        $ite(L = zero_zero_int,K,aa_int_int(gcd_gcd_int(L),aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9873_and__num_Oelims,axiom,
    ! [Xa: num,Xaa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Xa),Xaa) = Y )
     => ( ( ( Xa = one )
         => ( ( Xaa = one )
           => ( Y != some_num(one) ) ) )
       => ( ( ( Xa = one )
           => ( ? [N: num] : Xaa = aa_num_num(bit0,N)
             => ( Y != none_num ) ) )
         => ( ( ( Xa = one )
             => ( ? [N: num] : Xaa = aa_num_num(bit1,N)
               => ( Y != some_num(one) ) ) )
           => ( ( ? [M2: num] : Xa = aa_num_num(bit0,M2)
               => ( ( Xaa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M2: num] :
                    ( ( Xa = aa_num_num(bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M2),N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xa = aa_num_num(bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M2),N)) ) ) )
                 => ( ( ? [M2: num] : Xa = aa_num_num(bit1,M2)
                     => ( ( Xaa = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xa = aa_num_num(bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M2),N)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xa = aa_num_num(bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qq,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M2),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9874_xor__num_Oelims,axiom,
    ! [Xa: num,Xaa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Xa),Xaa) = Y )
     => ( ( ( Xa = one )
         => ( ( Xaa = one )
           => ( Y != none_num ) ) )
       => ( ( ( Xa = one )
           => ! [N: num] :
                ( ( Xaa = aa_num_num(bit0,N) )
               => ( Y != some_num(aa_num_num(bit1,N)) ) ) )
         => ( ( ( Xa = one )
             => ! [N: num] :
                  ( ( Xaa = aa_num_num(bit1,N) )
                 => ( Y != some_num(aa_num_num(bit0,N)) ) ) )
           => ( ! [M2: num] :
                  ( ( Xa = aa_num_num(bit0,M2) )
                 => ( ( Xaa = one )
                   => ( Y != some_num(aa_num_num(bit1,M2)) ) ) )
             => ( ! [M2: num] :
                    ( ( Xa = aa_num_num(bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M2),N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xa = aa_num_num(bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M2),N))) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xa = aa_num_num(bit1,M2) )
                       => ( ( Xaa = one )
                         => ( Y != some_num(aa_num_num(bit0,M2)) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xa = aa_num_num(bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M2),N))) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xa = aa_num_num(bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M2),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9875_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_9876_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_9877_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_9878_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_9879_gcd__0__nat,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),zero_zero_nat) = Xa ).

% gcd_0_nat
tff(fact_9880_gcd__0__left__nat,axiom,
    ! [Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Xa) = Xa ).

% gcd_0_left_nat
tff(fact_9881_gcd__Suc__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9882_gcd__pos__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),Nb))
    <=> ( ( Ma != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9883_gcd__red__nat,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)) ).

% gcd_red_nat
tff(fact_9884_gcd__non__0__nat,axiom,
    ! [Y: nat,Xa: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9885_gcd__nat_Osimps,axiom,
    ! [Xa: nat,Y: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y) = $ite(Y = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y))) ).

% gcd_nat.simps
tff(fact_9886_gcd__nat_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_9887_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9888_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_9889_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_9890_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_set_nat_o(member_nat(A4),A2)
         => ( aa_set_nat_o(member_nat(B4),A2)
           => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A4),B4)),A2) ) )
     => ( ( A2 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A2)),A2) ) ) ).

% Gcd_in
tff(fact_9891_and__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),one) = some_num(one) ).

% and_num.simps(1)
tff(fact_9892_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_9893_xor__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),one) = none_num ).

% xor_num.simps(1)
tff(fact_9894_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),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) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),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_9895_xor__num_Osimps_I5_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Ma),Nb)) ).

% xor_num.simps(5)
tff(fact_9896_and__num_Osimps_I5_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Ma),Nb)) ).

% and_num.simps(5)
tff(fact_9897_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      gcd_gcd_Code_integer(K,L) = abs_abs_Code_integer($ite(L = zero_z3403309356797280102nteger,K,gcd_gcd_Code_integer(L,modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))))) ).

% gcd_code_integer
tff(fact_9898_and__num_Osimps_I7_J,axiom,
    ! [Ma: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Ma)),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_9899_and__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit1,Nb)) = some_num(one) ).

% and_num.simps(3)
tff(fact_9900_and__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit0,Nb)) = none_num ).

% and_num.simps(2)
tff(fact_9901_and__num_Osimps_I4_J,axiom,
    ! [Ma: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Ma)),one) = none_num ).

% and_num.simps(4)
tff(fact_9902_and__num_Osimps_I8_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Ma),Nb)) ).

% and_num.simps(8)
tff(fact_9903_and__num_Osimps_I6_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Ma),Nb)) ).

% and_num.simps(6)
tff(fact_9904_xor__num_Osimps_I9_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Ma),Nb)) ).

% xor_num.simps(9)
tff(fact_9905_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_rf) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9906_xor__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit0,Nb)) = some_num(aa_num_num(bit1,Nb)) ).

% xor_num.simps(2)
tff(fact_9907_xor__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit1,Nb)) = some_num(aa_num_num(bit0,Nb)) ).

% xor_num.simps(3)
tff(fact_9908_xor__num_Osimps_I4_J,axiom,
    ! [Ma: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Ma)),one) = some_num(aa_num_num(bit1,Ma)) ).

% xor_num.simps(4)
tff(fact_9909_xor__num_Osimps_I7_J,axiom,
    ! [Ma: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Ma)),one) = some_num(aa_num_num(bit0,Ma)) ).

% xor_num.simps(7)
tff(fact_9910_and__num_Osimps_I9_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qq,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Ma),Nb)) ).

% and_num.simps(9)
tff(fact_9911_bezw__aux,axiom,
    ! [Xa: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(product_fst_int_int(bezw(Xa,Y))),aa_nat_int(semiri1314217659103216013at_int,Xa))),aa_int_int(times_times_int(product_snd_int_int(bezw(Xa,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9912_xor__num_Osimps_I8_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Ma),Nb))) ).

% xor_num.simps(8)
tff(fact_9913_xor__num_Osimps_I6_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Ma),Nb))) ).

% xor_num.simps(6)
tff(fact_9914_gcd__nat_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xaa) = Y )
     => ( accp_P4275260045618599050at_nat(gcd_nat_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))) )
           => ~ accp_P4275260045618599050at_nat(gcd_nat_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9915_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_9916_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_9917_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_rg,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9918_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9919_Inf__real__def,axiom,
    ! [X6: set_real] : comple4887499456419720421f_real(X6) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X6))) ).

% Inf_real_def
tff(fact_9920_Inf__nat__def1,axiom,
    ! [K6: set_nat] :
      ( ( K6 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K6)),K6) ) ).

% Inf_nat_def1
tff(fact_9921_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_9922_binomial__def,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_rh(Nb),K))) ).

% binomial_def
tff(fact_9923_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri,Nb),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ) ) ).

% range_mod
tff(fact_9924_suminf__eq__SUP__real,axiom,
    ! [X6: fun_nat_real] :
      ( summable_real(X6)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X6,I2))
       => ( suminf_real(X6) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_rj(X6),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9925_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9926_range__mult,axiom,
    ! [A: real] :
      image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9927_root__def,axiom,
    ! [Nb: nat,Xa: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xa) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_rk,Nb)),Xa)) ).

% root_def
tff(fact_9928_UNIV__bool,axiom,
    top_top_set_o = insert_o($false,insert_o($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_9929_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q2)
    <=> produc4947309494688390418_int_o(aTP_Lamm_rm(Q2),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9930_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q2)
    <=> produc4947309494688390418_int_o(aTP_Lamm_ro(Q2),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9931_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).

% card_UNIV_char
tff(fact_9932_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_9933_char_Osize_I2_J,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_size_char(char2((X15),(X2),(X32),(X42),(X52),(X62),(X7),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_9934_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% nat_of_char_less_256
tff(fact_9935_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_9936_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((B72))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B62)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B52)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B42)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B32)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B22)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B1)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_9937_char_Osize__gen,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_char(char2((X15),(X2),(X32),(X42),(X52),(X62),(X7),(X8))) = zero_zero_nat ).

% char.size_gen
tff(fact_9938_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_9939_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9940_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9941_upto__aux__rec,axiom,
    ! [Ia: int,J: int,Js: list_int] :
      upto_aux(Ia,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia),Js,upto_aux(Ia,aa_int_int(minus_minus_int(J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9942_upto_Opsimps,axiom,
    ! [Ia: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Ia,J))
     => ( upto(Ia,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J),cons_int(Ia,upto(aa_int_int(plus_plus_int(Ia),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9943_upto_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: list_int] :
      ( ( upto(Xa,Xaa) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Xa,Xaa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa),cons_int(Xa,upto(aa_int_int(plus_plus_int(Xa),one_one_int),Xaa)),nil_int) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Xa,Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9944_upto__empty,axiom,
    ! [J: int,Ia: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia)
     => ( upto(Ia,J) = nil_int ) ) ).

% upto_empty
tff(fact_9945_upto__Nil2,axiom,
    ! [Ia: int,J: int] :
      ( ( nil_int = upto(Ia,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia) ) ).

% upto_Nil2
tff(fact_9946_upto__Nil,axiom,
    ! [Ia: int,J: int] :
      ( ( upto(Ia,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),Ia) ) ).

% upto_Nil
tff(fact_9947_upto__single,axiom,
    ! [Ia: int] : upto(Ia,Ia) = cons_int(Ia,nil_int) ).

% upto_single
tff(fact_9948_nth__upto,axiom,
    ! [Ia: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(Ia),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(Ia,J)),K) = aa_int_int(plus_plus_int(Ia),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9949_length__upto,axiom,
    ! [Ia: int,J: int] : size_size_list_int(upto(Ia,J)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(J),Ia)),one_one_int)) ).

% length_upto
tff(fact_9950_upto__rec__numeral_I1_J,axiom,
    ! [Ma: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Ma),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Ma),upto(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Ma)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9951_upto__rec__numeral_I4_J,axiom,
    ! [Ma: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)),upto(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9952_upto__rec__numeral_I3_J,axiom,
    ! [Ma: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)),upto(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9953_upto__rec__numeral_I2_J,axiom,
    ! [Ma: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Ma),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Ma),upto(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Ma)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9954_upto__aux__def,axiom,
    ! [Ia: int,J: int,Js: list_int] : upto_aux(Ia,J,Js) = append_int(upto(Ia,J),Js) ).

% upto_aux_def
tff(fact_9955_upto__code,axiom,
    ! [Ia: int,J: int] : upto(Ia,J) = upto_aux(Ia,J,nil_int) ).

% upto_code
tff(fact_9956_atLeastAtMost__upto,axiom,
    ! [Ia: int,J: int] : set_or1266510415728281911st_int(Ia,J) = set_int2(upto(Ia,J)) ).

% atLeastAtMost_upto
tff(fact_9957_distinct__upto,axiom,
    ! [Ia: int,J: int] : aa_list_int_o(distinct_int,upto(Ia,J)) ).

% distinct_upto
tff(fact_9958_upto__split2,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(Ia,K) = append_int(upto(Ia,J),upto(aa_int_int(plus_plus_int(J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9959_upto__split1,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(Ia,K) = append_int(upto(Ia,aa_int_int(minus_minus_int(J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9960_atLeastLessThan__upto,axiom,
    ! [Ia: int,J: int] : set_or4662586982721622107an_int(Ia,J) = set_int2(upto(Ia,aa_int_int(minus_minus_int(J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9961_upto__rec1,axiom,
    ! [Ia: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( upto(Ia,J) = cons_int(Ia,upto(aa_int_int(plus_plus_int(Ia),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9962_upto_Osimps,axiom,
    ! [Ia: int,J: int] :
      upto(Ia,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J),cons_int(Ia,upto(aa_int_int(plus_plus_int(Ia),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9963_upto_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: list_int] :
      ( ( upto(Xa,Xaa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa),cons_int(Xa,upto(aa_int_int(plus_plus_int(Xa),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_9964_upto__rec2,axiom,
    ! [Ia: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( upto(Ia,J) = append_int(upto(Ia,aa_int_int(minus_minus_int(J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9965_upto__split3,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ia),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(Ia,K) = append_int(upto(Ia,aa_int_int(minus_minus_int(J),one_one_int)),cons_int(J,upto(aa_int_int(plus_plus_int(J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9966_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9967_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xa: real,D4: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xa != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D4,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9968_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9969_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9970_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_or1222579329274155063t_real(A,B))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X3)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9971_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9972_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9973_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H2))),aa_real_real(F,Xa)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9974_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H2))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9975_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H2))),aa_real_real(F,Xa)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9976_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H2))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9977_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9978_DERIV__isconst__all,axiom,
    ! [F: fun_real_real,Xa: real,Y: real] :
      ( ! [X3: real] : has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real))
     => ( aa_real_real(F,Xa) = aa_real_real(F,Y) ) ) ).

% DERIV_isconst_all
tff(fact_9979_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H2)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H2))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9980_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H2)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H2))),aa_real_real(F,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9981_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H2)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H2))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9982_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H2)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),H2))),aa_real_real(F,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9983_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y3))),D)
             => ( aa_real_real(F,Xa) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9984_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
       => ? [Z4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),B)
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F5,Z4)) ) ) ) ) ).

% MVT2
tff(fact_9985_DERIV__ln,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xa),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9986_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9987_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,Xa)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9988_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xa)),aa_real_real(F,Y3)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9989_DERIV__ln__divide,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xa),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9990_DERIV__pow,axiom,
    ! [Nb: nat,Xa: real,Sb: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_rp,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xa,Sb)) ).

% DERIV_pow
tff(fact_9991_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,Ma: real,Xa: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,Ma,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_rq(G),Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),Ma),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9992_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_rr,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9993_DERIV__log,axiom,
    ! [Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),Xa)),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9994_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Ma: real,Xa: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Ma,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xa))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_rs(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Ma),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9995_DERIV__powr,axiom,
    ! [G: fun_real_real,Ma: real,Xa: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Ma,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xa))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xa,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_rt(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xa)),aa_real_real(F,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,Xa)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ma),aa_real_real(F,Xa))),aa_real_real(G,Xa)))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9996_artanh__real__has__field__derivative,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xa,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9997_DERIV__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9998_DERIV__arctan,axiom,
    ! [Xa: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9999_arsinh__real__has__field__derivative,axiom,
    ! [Xa: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xa,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_10000_DERIV__real__sqrt__generic,axiom,
    ! [Xa: real,D4: real] :
      ( ( Xa != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
           => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_10001_arcosh__real__has__field__derivative,axiom,
    ! [Xa: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xa,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_10002_DERIV__real__root,axiom,
    ! [Nb: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_10003_DERIV__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_10004_DERIV__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_10005_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xa: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa))
          & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ru(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_10006_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xa: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ru(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_10007_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( Xa != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_10008_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T5: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rv(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_10009_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ? [T5: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rv(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_10010_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T5: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_rv(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_10011_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xa: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( Xa != zero_zero_real )
         => ( ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa))
                & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ru(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_10012_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xa: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ru(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_10013_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),B)
                 => ( ( Xa != C )
                   => ? [T5: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),T5)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T5)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xa) ) )
                        & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rw(Diff),C),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xa),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_10014_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T5: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),B)
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rx(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_10015_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T5: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),C)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rx(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_10016_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M2: nat,T5: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K) )
       => ! [M: nat,T6: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_rz(Nb),Diff),B2),M),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T6)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sa(Diff),M),T6),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_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,Nb),aa_nat_nat(suc,M)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_10017_DERIV__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_sb,suminf_real(aa_real_fun_nat_real(aTP_Lamm_sc,Xa)),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_10018_DERIV__power__series_H,axiom,
    ! [R3: real,F: fun_nat_real,X0: real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R3),R3))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_sd(F),X3)) )
     => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R3),R3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
         => has_fi5821293074295781190e_real(aTP_Lamm_sf(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_sd(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_10019_DERIV__isconst3,axiom,
    ! [A: real,B: real,Xa: real,Y: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_set_real_o(member_real(Xa),set_or1633881224788618240n_real(A,B))
       => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A,B))
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) )
           => ( aa_real_real(F,Xa) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_10020_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F5: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L4: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_sg(F),N),aa_nat_real(aa_real_fun_nat_real(F5,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X3)) )
       => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F5,X0))
           => ( summable_real(L4)
             => ( ! [N: nat,X3: real,Y3: real] :
                    ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(A,B))
                   => ( aa_set_real_o(member_real(Y3),set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X3),N)),aa_nat_real(aa_real_fun_nat_real(F,Y3),N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L4,N)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),Y3)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_sh(F),suminf_real(aa_real_fun_nat_real(F5,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_10021_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_10022_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_10023_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
            & ! [X: real] :
                ( ( ( X != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X))),R) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_10024_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
            & ! [X: real] :
                ( ( ( X != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X))),R) )
               => ( aa_real_real(F,X) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_10025_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
            & ! [X: real] :
                ( ( ( X != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X))),R) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_10026_isCont__ln,axiom,
    ! [Xa: real] :
      ( ( Xa != zero_zero_real )
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),ln_ln_real) ) ).

% isCont_ln
tff(fact_10027_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ia)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(Ia),J)) = cons_nat(aa_nat_nat(suc,Ia),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,Ia)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_10028_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_si,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_10029_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,Ia)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(Ia),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_10030_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D4: real,G: fun_real_real,Xa: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D4,topolo2177554685111907308n_real(aa_real_real(G,Xa),top_top_set_real))
     => ( ( D4 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
           => ( ! [Y3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),B)
                   => ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D4),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_10031_LIM__less__bound,axiom,
    ! [B: real,Xa: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xa)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(B,Xa))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xa,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xa)) ) ) ) ).

% LIM_less_bound
tff(fact_10032_greaterThanLessThan__upto,axiom,
    ! [Ia: int,J: int] : set_or5832277885323065728an_int(Ia,J) = set_int2(upto(aa_int_int(plus_plus_int(Ia),one_one_int),aa_int_int(minus_minus_int(J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_10033_isCont__inverse__function,axiom,
    ! [D: real,Xa: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z4),Xa))),D)
           => ( aa_real_real(G,aa_real_real(F,Z4)) = Z4 ) )
       => ( ! [Z4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z4),Xa))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z4,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xa),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_10034_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z4),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z4,top_top_set_real),F) ) )
       => ( ! [Z4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z4),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z4,top_top_set_real),G) ) )
         => ( ! [Z4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),B)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z4),topolo2177554685111907308n_real(Z4,top_top_set_real)) ) )
           => ( ! [Z4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z4)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F5,Z4),topolo2177554685111907308n_real(Z4,top_top_set_real)) ) )
             => ? [C2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F5,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_10035_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N5: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_sj(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_sj(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))),N5)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_sj(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))),N5))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_10036_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N5: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_sj(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_sj(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))),N5))),groups6591440286371151544t_real(aTP_Lamm_sj(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))),N5)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_10037_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_10038_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_10039_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_10040_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_av,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_10041_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sk(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N5)),L2)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_nat_real(G,N5))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_10042_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R: real] :
        ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R),aa_nat_real(X6,N)) )
     => filterlim_nat_real(aTP_Lamm_sl(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_10043_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_sm,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_10044_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_10045_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_so,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_10046_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sp,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_10047_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
             => ? [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N5)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_10048_LIMSEQ__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_10049_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sq(Xa),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_10050_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_10051_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_10052_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sr,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_10053_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ss,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10054_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_st,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10055_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_sj(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_10056_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_sj(A)) ) ) ) ).

% summable
tff(fact_10057_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_su(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_sv(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_10058_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_sw(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_sv(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_10059_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_sx(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sj(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_10060_zeroseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_oc,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_10061_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_sx(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sj(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_10062_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_sj(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),suminf_real(aTP_Lamm_sj(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_10063_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_sj(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))),N5)))),L2)
              & filterlim_nat_real(aTP_Lamm_sx(A),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),groups6591440286371151544t_real(aTP_Lamm_sj(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))),N5)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_sy(A),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_10064_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_sy(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sj(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_10065_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_sj(A))),groups6591440286371151544t_real(aTP_Lamm_sj(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_10066_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_sy(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sj(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_10067_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C4: real] :
        ! [X4: real] : aa_real_real(F,X4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X4),C4) ) ).

% real_bounded_linear
tff(fact_10068_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_sz,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_10069_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_ta,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10070_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10071_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X3)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_10072_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10073_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tb(Nb),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10074_exp__at__bot,axiom,
    filterlim_real_real(exp_real,topolo2815343760600316023s_real(zero_zero_real),at_bot_real) ).

% exp_at_bot
tff(fact_10075_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_10076_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_10077_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tb(Nb),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10078_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_10079_tendsto__exp__limit__at__right,axiom,
    ! [Xa: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_tc,Xa),topolo2815343760600316023s_real(aa_real_real(exp_real,Xa)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_10080_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_10081_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_10082_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_10083_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real(arcosh_real,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(one_one_real,set_or5849166863359141190n_real(one_one_real))) ).

% tendsto_arcosh_at_left_1
tff(fact_10084_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_10085_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xa: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
     => ( eventually_real(aTP_Lamm_td(G2),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F),F5),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G),G2),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_10086_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_10087_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_10088_eventually__at__right__to__0,axiom,
    ! [P: fun_real_o,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_real_fun_real_o(aTP_Lamm_tg(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_10089_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_10090_eventually__at__right__to__top,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_th(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_10091_eventually__at__top__to__right,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_th(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_10092_lhopital,axiom,
    ! [F: fun_real_real,Xa: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xa,top_top_set_real))
       => ( eventually_real(aTP_Lamm_td(G),topolo2177554685111907308n_real(Xa,top_top_set_real))
         => ( eventually_real(aTP_Lamm_td(G2),topolo2177554685111907308n_real(Xa,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F),F5),topolo2177554685111907308n_real(Xa,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G),G2),topolo2177554685111907308n_real(Xa,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),F3,topolo2177554685111907308n_real(Xa,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),F3,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_10093_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Xa: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_td(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F),F5),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),topolo2815343760600316023s_real(Xa),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G),F),topolo2815343760600316023s_real(Xa),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_10094_lhopital__at__top,axiom,
    ! [G: fun_real_real,Xa: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( eventually_real(aTP_Lamm_td(G2),topolo2177554685111907308n_real(Xa,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F),F5),topolo2177554685111907308n_real(Xa,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G),G2),topolo2177554685111907308n_real(Xa,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xa,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_10095_lhopital__right,axiom,
    ! [F: fun_real_real,Xa: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
       => ( eventually_real(aTP_Lamm_td(G),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
         => ( eventually_real(aTP_Lamm_td(G2),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F),F5),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G),G2),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),F3,topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),F3,topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_10096_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_td(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_td(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F0),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_10097_lhopital__left,axiom,
    ! [F: fun_real_real,Xa: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
       => ( eventually_real(aTP_Lamm_td(G),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
         => ( eventually_real(aTP_Lamm_td(G2),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F),F5),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G),G2),topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),F3,topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(F),G),F3,topolo2177554685111907308n_real(Xa,set_or5984915006950818249n_real(Xa))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_10098_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Xa: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_td(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),topolo2815343760600316023s_real(Xa),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G),F),topolo2815343760600316023s_real(Xa),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_10099_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,Xa: real,G2: fun_real_real,F: fun_real_real,F5: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
     => ( eventually_real(aTP_Lamm_td(G2),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(F),F5),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_te(G),G2),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G2),F5),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xa,set_or5849166863359141190n_real(Xa))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_10100_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_tj(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10101_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_tk(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10102_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_tk(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_10103_Bseq__realpow,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xa),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_10104_Gcd__eq__Max,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( ( M6 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_tl,M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_10105_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dz,Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_10106_Sup__nat__def,axiom,
    ! [X6: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X6) = $ite(X6 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X6)) ).

% Sup_nat_def
tff(fact_10107_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2))) ) ).

% card_le_Suc_Max
tff(fact_10108_divide__nat__def,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_tm(Ma),Nb)))) ).

% divide_nat_def
tff(fact_10109_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),Nb) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_tn(Nb),Ma))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_10110_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ed,Nb))) = aa_int_int(abs_abs_int,Nb) ) ) ).

% Max_divisors_self_int
tff(fact_10111_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_10112_gcd__is__Max__divisors__int,axiom,
    ! [Nb: int,Ma: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_int(gcd_gcd_int(Ma),Nb) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_to(Nb),Ma))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_10113_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ia)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(Ia),J)) = cons_nat(aa_nat_nat(suc,Ia),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,Ia)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_10114_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(Ia),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_10115_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg,TreeList2,Summary2) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary2),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) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_tp(Deg),TreeList2),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_10116_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_10117_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_10118_greaterThanAtMost__upto,axiom,
    ! [Ia: int,J: int] : set_or6656581121297822940st_int(Ia,J) = set_int2(upto(aa_int_int(plus_plus_int(Ia),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_10119_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_10120_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg2: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg2)
    <=> ( ( Dega = Deg2 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n),
              ( ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X4),n) )
              & aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
              & ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X9)
                    & ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_tp(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_10121_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg,TreeList2,Summary2) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary2),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) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                            & ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                               => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_tp(Deg),TreeList2),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_10122_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg,TreeList2,Summary2) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary2),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) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                              & ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_tp(Deg),TreeList2),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_10123_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg,TreeList2,Summary2) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary2),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) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                     => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_tp(Deg),TreeList2),Summary2),n),m2))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList2,Summary2),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_10124_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList2,Summary2),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary2),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) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                     => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_tp(Deg),TreeList2),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_10125_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg,TreeList2,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList2,Summary2),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary2),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) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X9)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_tp(Deg),TreeList2),Summary2),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_10126_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
       => ( ! [X3: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) )
         => ( ! [X3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),G) )
           => ( ! [X3: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X3,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_10127_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ? [L2: real,Z4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),B)
              & has_fi5821293074295781190e_real(F,L2,topolo2177554685111907308n_real(Z4,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),L2) ) ) ) ) ) ).

% MVT
tff(fact_10128_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),B)
                & ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(F5,Z4),X) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_10129_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_10130_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_10131_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_10132_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),B)
             => ( aa_real_real(F,Xa) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_10133_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),B)
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z4,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_10134_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_tr,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_10135_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_tt,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_10136_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10137_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_10138_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tu,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_10139_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_rk,Nb)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_10140_log__inj,axiom,
    ! [B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => aa_set_real_o(inj_on_real_real(aa_rea7340605660300375871l_real(log,B)),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_10141_inj__Suc,axiom,
    ! [N3: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N3) ).

% inj_Suc
tff(fact_10142_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qk,K)),N3) ) ).

% inj_on_diff_nat
tff(fact_10143_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_10144_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F)) ) ) ) ).

% suminf_reindex_mono
tff(fact_10145_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% inj_on_char_of_nat
tff(fact_10146_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
         => ( ! [X3: nat] :
                ( ~ aa_set_nat_o(member_nat(X3),image_nat_nat(G,top_top_set_nat))
               => ( aa_nat_real(F,X3) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_10147_min__Suc__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Ma),Nb)) ).

% min_Suc_Suc
tff(fact_10148_min__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% min_0L
tff(fact_10149_min__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_10150_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_10151_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% min_numeral_Suc
tff(fact_10152_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_10153_concat__bit__assoc__sym,axiom,
    ! [Ma: nat,Nb: nat,K: int,L: int,R2: int] : aa_int_int(bit_concat_bit(Ma,aa_int_int(bit_concat_bit(Nb,K),L)),R2) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Ma),Nb),K),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_10154_nat__mult__min__right,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Q2)) ).

% nat_mult_min_right
tff(fact_10155_nat__mult__min__left,axiom,
    ! [Ma: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Ma),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_min_left
tff(fact_10156_min__diff,axiom,
    ! [Ma: nat,Ia: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Ia)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ia)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Ma),Nb)),Ia) ).

% min_diff
tff(fact_10157_take__bit__concat__bit__eq,axiom,
    ! [Ma: nat,Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Ma),aa_int_int(bit_concat_bit(Nb,K),L)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Ma),Nb),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),L)) ).

% take_bit_concat_bit_eq
tff(fact_10158_min__Suc1,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),Ma) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_tv,Nb)),Ma) ).

% min_Suc1
tff(fact_10159_min__Suc2,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_tw,Nb)),Ma) ).

% min_Suc2
tff(fact_10160_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,zero_z5237406670263579293d_enat),Q2) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_10161_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q2),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_10162_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
tff(fact_10163_powr__real__of__int_H,axiom,
    ! [Xa: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( ( Xa != zero_zero_real )
          | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xa),aa_int_real(ring_1_of_int_real,Nb)) = aa_int_real(power_int_real(Xa),Nb) ) ) ) ).

% powr_real_of_int'
tff(fact_10164_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_tx)) ).

% pred_nat_def
tff(fact_10165_less__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ma),Nb),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% less_eq
tff(fact_10166_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_ty) ).

% Rats_eq_int_div_nat
tff(fact_10167_Rats__abs__iff,axiom,
    ! [Xa: real] :
      ( aa_set_real_o(member_real(aa_real_real(abs_abs_real,Xa)),field_5140801741446780682s_real)
    <=> aa_set_real_o(member_real(Xa),field_5140801741446780682s_real) ) ).

% Rats_abs_iff
tff(fact_10168_Rats__no__top__le,axiom,
    ! [Xa: real] :
    ? [X3: real] :
      ( aa_set_real_o(member_real(X3),field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X3) ) ).

% Rats_no_top_le
tff(fact_10169_Rats__no__bot__less,axiom,
    ! [Xa: real] :
    ? [X3: real] :
      ( aa_set_real_o(member_real(X3),field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Xa) ) ).

% Rats_no_bot_less
tff(fact_10170_Rats__dense__in__real,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ? [X3: real] :
          ( aa_set_real_o(member_real(X3),field_5140801741446780682s_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y) ) ) ).

% Rats_dense_in_real
tff(fact_10171_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_tz) ).

% Rats_eq_int_div_int
tff(fact_10172_complex__is__Nat__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),semiri3842193898606819883omplex)
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I: nat] : aa_complex_real(re,Z) = aa_nat_real(semiri5074537144036343181t_real,I) ) ) ).

% complex_is_Nat_iff
tff(fact_10173_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F5: fun_real_real] :
      ( ! [X3: real] : has_fi5821293074295781190e_real(F,aa_real_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real))
     => ( ! [X3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F5,X3))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_10174_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(F,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_10175_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( ! [X3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_ua(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_10176_upt__rec__numeral,axiom,
    ! [Ma: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,Ma),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Ma),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_10177_remdups__upt,axiom,
    ! [Ma: nat,Nb: nat] : remdups_nat(upt(Ma,Nb)) = upt(Ma,Nb) ).

% remdups_upt
tff(fact_10178_tl__upt,axiom,
    ! [Ma: nat,Nb: nat] : tl_nat(upt(Ma,Nb)) = upt(aa_nat_nat(suc,Ma),Nb) ).

% tl_upt
tff(fact_10179_hd__upt,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( hd_nat(upt(Ia,J)) = Ia ) ) ).

% hd_upt
tff(fact_10180_drop__upt,axiom,
    ! [Ma: nat,Ia: nat,J: nat] : drop_nat(Ma,upt(Ia,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Ma),J) ).

% drop_upt
tff(fact_10181_length__upt,axiom,
    ! [Ia: nat,J: nat] : size_size_list_nat(upt(Ia,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Ia) ).

% length_upt
tff(fact_10182_take__upt,axiom,
    ! [Ia: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Ma)),Nb)
     => ( take_nat(Ma,upt(Ia,Nb)) = upt(Ia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),Ma)) ) ) ).

% take_upt
tff(fact_10183_upt__conv__Nil,axiom,
    ! [J: nat,Ia: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia)
     => ( upt(Ia,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_10184_sorted__list__of__set__range,axiom,
    ! [Ma: nat,Nb: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(Ma),Nb)) = upt(Ma,Nb) ).

% sorted_list_of_set_range
tff(fact_10185_upt__eq__Nil__conv,axiom,
    ! [Ia: nat,J: nat] :
      ( ( upt(Ia,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Ia) ) ) ).

% upt_eq_Nil_conv
tff(fact_10186_nth__upt,axiom,
    ! [Ia: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K)),J)
     => ( aa_nat_nat(nth_nat(upt(Ia,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),K) ) ) ).

% nth_upt
tff(fact_10187_upt__conv__Cons,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
     => ( upt(Ia,J) = cons_nat(Ia,upt(aa_nat_nat(suc,Ia),J)) ) ) ).

% upt_conv_Cons
tff(fact_10188_atMost__upto,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).

% atMost_upto
tff(fact_10189_upt__conv__Cons__Cons,axiom,
    ! [Ma: nat,Nb: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(Ma,cons_nat(Nb,Ns)) = upt(Ma,Q2) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Ma),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_10190_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(Nb),Ma) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Ma))) ).

% greaterThanAtMost_upt
tff(fact_10191_atLeast__upt,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).

% atLeast_upt
tff(fact_10192_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(Nb),Ma) = set_nat2(upt(aa_nat_nat(suc,Nb),Ma)) ).

% greaterThanLessThan_upt
tff(fact_10193_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Ma) = set_nat2(upt(Nb,aa_nat_nat(suc,Ma))) ).

% atLeastAtMost_upt
tff(fact_10194_atLeastLessThan__upt,axiom,
    ! [Ia: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Ia),J) = set_nat2(upt(Ia,J)) ).

% atLeastLessThan_upt
tff(fact_10195_distinct__upt,axiom,
    ! [Ia: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(Ia,J)) ).

% distinct_upt
tff(fact_10196_upt__0,axiom,
    ! [Ia: nat] : upt(Ia,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_10197_upt__add__eq__append,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( upt(Ia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(Ia,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_10198_upt__eq__Cons__conv,axiom,
    ! [Ia: nat,J: nat,Xa: nat,Xs: list_nat] :
      ( ( upt(Ia,J) = cons_nat(Xa,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J)
        & ( Ia = Xa )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ia),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_10199_upt__rec,axiom,
    ! [Ia: nat,J: nat] :
      upt(Ia,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),J),cons_nat(Ia,upt(aa_nat_nat(suc,Ia),J)),nil_nat) ).

% upt_rec
tff(fact_10200_upt__Suc__append,axiom,
    ! [Ia: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J)
     => ( upt(Ia,aa_nat_nat(suc,J)) = append_nat(upt(Ia,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_10201_upt__Suc,axiom,
    ! [Ia: nat,J: nat] :
      upt(Ia,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),J),append_nat(upt(Ia,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_10202_map__add__upt,axiom,
    ! [Nb: nat,Ma: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ub,Nb),upt(zero_zero_nat,Ma)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% map_add_upt
tff(fact_10203_map__Suc__upt,axiom,
    ! [Ma: nat,Nb: nat] : map_nat_nat(suc,upt(Ma,Nb)) = upt(aa_nat_nat(suc,Ma),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_10204_map__decr__upt,axiom,
    ! [Ma: nat,Nb: nat] : map_nat_nat(aTP_Lamm_uc,upt(aa_nat_nat(suc,Ma),aa_nat_nat(suc,Nb))) = upt(Ma,Nb) ).

% map_decr_upt
tff(fact_10205_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_ud,Qr) ).

% Divides.adjust_div_def
tff(fact_10206_card__length__sum__list__rec,axiom,
    ! [Ma: nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Ma)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ue(Ma),N3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_uf(Ma),N3)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ug(Ma),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_10207_card__length__sum__list,axiom,
    ! [Ma: nat,N3: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ue(Ma),N3))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),Ma)),one_one_nat)),N3) ).

% card_length_sum_list
tff(fact_10208_sorted__wrt__upt,axiom,
    ! [Ma: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Ma,Nb)) ).

% sorted_wrt_upt
tff(fact_10209_sorted__upt,axiom,
    ! [Ma: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Ma,Nb)) ).

% sorted_upt
tff(fact_10210_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,Ia: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ia),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ia),aa_nat_nat(nth_nat(Ns),Ia)) ) ) ).

% sorted_wrt_less_idx
tff(fact_10211_sorted__upto,axiom,
    ! [Ma: int,Nb: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(Ma,Nb)) ).

% sorted_upto
tff(fact_10212_sorted__wrt__upto,axiom,
    ! [Ia: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(Ia,J)) ).

% sorted_wrt_upto
tff(fact_10213_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_of,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10214_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_oq,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10215_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_pi,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10216_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_oj,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10217_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_si,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10218_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_sz,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10219_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_uc,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_7
tff(fact_10220_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_sm,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_8
tff(fact_10221_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_sb,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_oe,Uu)) ).

% ATP.lambda_9
tff(fact_10222_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_tl,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_dz,Uu)) ).

% ATP.lambda_10
tff(fact_10223_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_tt,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_ts(Uu)))) ).

% ATP.lambda_11
tff(fact_10224_ATP_Olambda__12,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_tr,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_tq(Uu)))) ).

% ATP.lambda_12
tff(fact_10225_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_so,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_13
tff(fact_10226_ATP_Olambda__14,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_qq,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_14
tff(fact_10227_ATP_Olambda__15,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_qp,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_15
tff(fact_10228_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_gs,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_16
tff(fact_10229_ATP_Olambda__17,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_ty,Uu)
    <=> ? [I: int,N4: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_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_17
tff(fact_10230_ATP_Olambda__18,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_tz,Uu)
    <=> ? [I: int,J2: int] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I)),aa_int_real(ring_1_of_int_real,J2)) )
          & ( J2 != zero_zero_int ) ) ) ).

% ATP.lambda_18
tff(fact_10231_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_qo,Uu) = some_num(one) ).

% ATP.lambda_19
tff(fact_10232_ATP_Olambda__20,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_rd(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_rb(Uua),aTP_Lamm_rc(Uua),Uu) ).

% ATP.lambda_20
tff(fact_10233_ATP_Olambda__21,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_or(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_21
tff(fact_10234_ATP_Olambda__22,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qn,Uu),Uua) = $ite(Uu = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,Uu)),Uua),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_22
tff(fact_10235_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_re,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_rd(Uua),Uu) ).

% ATP.lambda_23
tff(fact_10236_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_rb(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_qp,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_24
tff(fact_10237_ATP_Olambda__25,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_qr(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_qp,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_25
tff(fact_10238_ATP_Olambda__26,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_od,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_26
tff(fact_10239_ATP_Olambda__27,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_op,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_27
tff(fact_10240_ATP_Olambda__28,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ot,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(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_28
tff(fact_10241_ATP_Olambda__29,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_pj,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_29
tff(fact_10242_ATP_Olambda__30,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_pg,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_30
tff(fact_10243_ATP_Olambda__31,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_pz,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(plus_plus_int(l2),one_one_int)) ) ).

% ATP.lambda_31
tff(fact_10244_ATP_Olambda__32,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_pm(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_32
tff(fact_10245_ATP_Olambda__33,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_pl(Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Uua),one_one_int))) ) ) ).

% ATP.lambda_33
tff(fact_10246_ATP_Olambda__34,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oe,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_34
tff(fact_10247_ATP_Olambda__35,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sc,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_35
tff(fact_10248_ATP_Olambda__36,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sj(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_36
tff(fact_10249_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_rf,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_37
tff(fact_10250_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pd,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_38
tff(fact_10251_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ol,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_39
tff(fact_10252_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ta,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_40
tff(fact_10253_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tu,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_41
tff(fact_10254_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ow,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_42
tff(fact_10255_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ov,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_43
tff(fact_10256_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ec(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_44
tff(fact_10257_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_eb(Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_45
tff(fact_10258_ATP_Olambda__46,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_tc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_46
tff(fact_10259_ATP_Olambda__47,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_47
tff(fact_10260_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ph,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_48
tff(fact_10261_ATP_Olambda__49,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oa(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_49
tff(fact_10262_ATP_Olambda__50,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nx(Uu),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_50
tff(fact_10263_ATP_Olambda__51,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nz(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_51
tff(fact_10264_ATP_Olambda__52,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ny(Uu),Uua) = aa_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_52
tff(fact_10265_ATP_Olambda__53,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_np(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_53
tff(fact_10266_ATP_Olambda__54,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nn(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_54
tff(fact_10267_ATP_Olambda__55,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_no(Uu),Uua) = aa_int_int(minus_minus_int(aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_55
tff(fact_10268_ATP_Olambda__56,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mv(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_56
tff(fact_10269_ATP_Olambda__57,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mx(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_57
tff(fact_10270_ATP_Olambda__58,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mw(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_58
tff(fact_10271_ATP_Olambda__59,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nw(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_59
tff(fact_10272_ATP_Olambda__60,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nu(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_60
tff(fact_10273_ATP_Olambda__61,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nv(Uu),Uua) = aa_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_61
tff(fact_10274_ATP_Olambda__62,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_nq(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_62
tff(fact_10275_ATP_Olambda__63,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_63
tff(fact_10276_ATP_Olambda__64,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nr(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_64
tff(fact_10277_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ns(Uu),Uua) = aa_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_65
tff(fact_10278_ATP_Olambda__66,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_kh(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_66
tff(fact_10279_ATP_Olambda__67,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_kk(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_67
tff(fact_10280_ATP_Olambda__68,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_kd(Uu),Uua)
    <=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_68
tff(fact_10281_ATP_Olambda__69,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_kf(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_69
tff(fact_10282_ATP_Olambda__70,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( aa_real_o(aTP_Lamm_ki(Uu),Uua)
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_70
tff(fact_10283_ATP_Olambda__71,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_kl(Uu),Uua)
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_71
tff(fact_10284_ATP_Olambda__72,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( aa_int_o(aTP_Lamm_ke(Uu),Uua)
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_72
tff(fact_10285_ATP_Olambda__73,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_kg(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_73
tff(fact_10286_ATP_Olambda__74,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_kj(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_74
tff(fact_10287_ATP_Olambda__75,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_km(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_75
tff(fact_10288_ATP_Olambda__76,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sy(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_sj(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_76
tff(fact_10289_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sx(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_sj(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_77
tff(fact_10290_ATP_Olambda__78,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_qm(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_78
tff(fact_10291_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rk,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_79
tff(fact_10292_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oo,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_80
tff(fact_10293_ATP_Olambda__81,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_om,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_81
tff(fact_10294_ATP_Olambda__82,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_rg,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).

% ATP.lambda_82
tff(fact_10295_ATP_Olambda__83,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_py,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_83
tff(fact_10296_ATP_Olambda__84,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gl,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_84
tff(fact_10297_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gk,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_85
tff(fact_10298_ATP_Olambda__86,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gj,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_86
tff(fact_10299_ATP_Olambda__87,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_go,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_87
tff(fact_10300_ATP_Olambda__88,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gn,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_88
tff(fact_10301_ATP_Olambda__89,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gm,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_89
tff(fact_10302_ATP_Olambda__90,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pe,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_90
tff(fact_10303_ATP_Olambda__91,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_st,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_91
tff(fact_10304_ATP_Olambda__92,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ud,Uu),Uua) = aa_int_int(plus_plus_int(Uu),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_92
tff(fact_10305_ATP_Olambda__93,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ss,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_93
tff(fact_10306_ATP_Olambda__94,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_94
tff(fact_10307_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rj(Uu),Uua) = groups6591440286371151544t_real(Uu,aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_95
tff(fact_10308_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_tx,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_96
tff(fact_10309_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_po,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_97
tff(fact_10310_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_98
tff(fact_10311_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ea,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_99
tff(fact_10312_ATP_Olambda__100,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_at,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).

% ATP.lambda_100
tff(fact_10313_ATP_Olambda__101,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_au(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_101
tff(fact_10314_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_av,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_102
tff(fact_10315_ATP_Olambda__103,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_aw(Uu),Uua) = aa_int_int(times_times_int(Uua),Uu) ).

% ATP.lambda_103
tff(fact_10316_ATP_Olambda__104,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qk,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_104
tff(fact_10317_ATP_Olambda__105,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rp,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_105
tff(fact_10318_ATP_Olambda__106,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ub,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_106
tff(fact_10319_ATP_Olambda__107,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_ql(Uu),Uua) = aa_int_int(plus_plus_int(Uua),Uu) ).

% ATP.lambda_107
tff(fact_10320_ATP_Olambda__108,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_rr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_108
tff(fact_10321_ATP_Olambda__109,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dz,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_109
tff(fact_10322_ATP_Olambda__110,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ed,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_110
tff(fact_10323_ATP_Olambda__111,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ou,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_111
tff(fact_10324_ATP_Olambda__112,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_112
tff(fact_10325_ATP_Olambda__113,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_113
tff(fact_10326_ATP_Olambda__114,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aa(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_114
tff(fact_10327_ATP_Olambda__115,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_115
tff(fact_10328_ATP_Olambda__116,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_116
tff(fact_10329_ATP_Olambda__117,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_117
tff(fact_10330_ATP_Olambda__118,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_118
tff(fact_10331_ATP_Olambda__119,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aj,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_119
tff(fact_10332_ATP_Olambda__120,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ah(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_120
tff(fact_10333_ATP_Olambda__121,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_ak((Uu)),(Uua))
    <=> ( (Uua)
      <=> (Uu) ) ) ).

% ATP.lambda_121
tff(fact_10334_ATP_Olambda__122,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ai(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_122
tff(fact_10335_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_al,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_123
tff(fact_10336_ATP_Olambda__124,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_am,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_124
tff(fact_10337_ATP_Olambda__125,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(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_125
tff(fact_10338_ATP_Olambda__126,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oh(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_126
tff(fact_10339_ATP_Olambda__127,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_th(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(inverse_inverse_real,Uua)) ) ).

% ATP.lambda_127
tff(fact_10340_ATP_Olambda__128,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nc(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_128
tff(fact_10341_ATP_Olambda__129,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_tj(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_129
tff(fact_10342_ATP_Olambda__130,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nl(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_130
tff(fact_10343_ATP_Olambda__131,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nb(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_131
tff(fact_10344_ATP_Olambda__132,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_nm(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_132
tff(fact_10345_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_rc(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_133
tff(fact_10346_ATP_Olambda__134,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_qs(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_134
tff(fact_10347_ATP_Olambda__135,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qj,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qi(Uu),Uua)) ).

% ATP.lambda_135
tff(fact_10348_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qh,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qg(Uu),Uua)) ).

% ATP.lambda_136
tff(fact_10349_ATP_Olambda__137,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_qf,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_qe(Uu),Uua)) ).

% ATP.lambda_137
tff(fact_10350_ATP_Olambda__138,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_qd,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_qc(Uu),Uua)) ).

% ATP.lambda_138
tff(fact_10351_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qb,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qa(Uu),Uua)) ).

% ATP.lambda_139
tff(fact_10352_ATP_Olambda__140,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_sf(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_se(Uu),Uua)) ).

% ATP.lambda_140
tff(fact_10353_ATP_Olambda__141,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_pf(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_141
tff(fact_10354_ATP_Olambda__142,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_td(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_142
tff(fact_10355_ATP_Olambda__143,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gt,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_143
tff(fact_10356_ATP_Olambda__144,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sr,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_144
tff(fact_10357_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tv,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_145
tff(fact_10358_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tw,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_146
tff(fact_10359_ATP_Olambda__147,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pt,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_147
tff(fact_10360_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ps,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_148
tff(fact_10361_ATP_Olambda__149,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_fy(Uu),Uua)
    <=> ~ aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_149
tff(fact_10362_ATP_Olambda__150,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_fz(Uu),Uua)
    <=> ~ aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_150
tff(fact_10363_ATP_Olambda__151,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ga(Uu),Uua)
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_151
tff(fact_10364_ATP_Olambda__152,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_gb(Uu),Uua)
    <=> ~ aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_152
tff(fact_10365_ATP_Olambda__153,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_gc(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_153
tff(fact_10366_ATP_Olambda__154,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_gd(Uu),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_10367_ATP_Olambda__155,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sl(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_155
tff(fact_10368_ATP_Olambda__156,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hq(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_156
tff(fact_10369_ATP_Olambda__157,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_hp(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_157
tff(fact_10370_ATP_Olambda__158,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sw(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_158
tff(fact_10371_ATP_Olambda__159,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_sh(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_159
tff(fact_10372_ATP_Olambda__160,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_ge(Uu),Uua)
    <=> ~ aa_real_o(Uu,Uua) ) ).

% ATP.lambda_160
tff(fact_10373_ATP_Olambda__161,axiom,
    ! [Uu: fun_list_nat_o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_gf(Uu),Uua)
    <=> ~ aa_list_nat_o(Uu,Uua) ) ).

% ATP.lambda_161
tff(fact_10374_ATP_Olambda__162,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_gg(Uu),Uua)
    <=> ~ aa_set_nat_o(Uu,Uua) ) ).

% ATP.lambda_162
tff(fact_10375_ATP_Olambda__163,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_gh(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_163
tff(fact_10376_ATP_Olambda__164,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_gi(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_164
tff(fact_10377_ATP_Olambda__165,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_os(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_165
tff(fact_10378_ATP_Olambda__166,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_og(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_166
tff(fact_10379_ATP_Olambda__167,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gr(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_167
tff(fact_10380_ATP_Olambda__168,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gp(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_168
tff(fact_10381_ATP_Olambda__169,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gq(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(minus_minus_int(Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_169
tff(fact_10382_ATP_Olambda__170,axiom,
    ! [Uu: fun_complex_real,Uua: set_complex,Uub: complex] :
      aa_complex_real(aa_set904117813986905849x_real(aTP_Lamm_jx(Uu),Uua),Uub) = $ite(aa_set_complex_o(member_complex(Uub),Uua),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_170
tff(fact_10383_ATP_Olambda__171,axiom,
    ! [Uu: fun_complex_rat,Uua: set_complex,Uub: complex] :
      aa_complex_rat(aa_set7482584248617139493ex_rat(aTP_Lamm_ka(Uu),Uua),Uub) = $ite(aa_set_complex_o(member_complex(Uub),Uua),aa_complex_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_171
tff(fact_10384_ATP_Olambda__172,axiom,
    ! [Uu: fun_real_complex,Uua: set_real,Uub: real] :
      aa_real_complex(aa_set2225337358267413879omplex(aTP_Lamm_jt(Uu),Uua),Uub) = $ite(aa_set_real_o(member_real(Uub),Uua),aa_real_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_172
tff(fact_10385_ATP_Olambda__173,axiom,
    ! [Uu: fun_real_real,Uua: set_real,Uub: real] :
      aa_real_real(aa_set8744029839670122869l_real(aTP_Lamm_jv(Uu),Uua),Uub) = $ite(aa_set_real_o(member_real(Uub),Uua),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_173
tff(fact_10386_ATP_Olambda__174,axiom,
    ! [Uu: fun_real_rat,Uua: set_real,Uub: real] :
      aa_real_rat(aa_set4110029769364456865al_rat(aTP_Lamm_jy(Uu),Uua),Uub) = $ite(aa_set_real_o(member_real(Uub),Uua),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_174
tff(fact_10387_ATP_Olambda__175,axiom,
    ! [Uu: fun_real_nat,Uua: set_real,Uub: real] :
      aa_real_nat(aa_set7618258327757853593al_nat(aTP_Lamm_kb(Uu),Uua),Uub) = $ite(aa_set_real_o(member_real(Uub),Uua),aa_real_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_175
tff(fact_10388_ATP_Olambda__176,axiom,
    ! [Uu: fun_int_complex,Uua: set_int,Uub: int] :
      aa_int_complex(aa_set2721330520047917815omplex(aTP_Lamm_ju(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_176
tff(fact_10389_ATP_Olambda__177,axiom,
    ! [Uu: fun_int_real,Uua: set_int,Uub: int] :
      aa_int_real(aa_set6366552415685541621t_real(aTP_Lamm_jw(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_177
tff(fact_10390_ATP_Olambda__178,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_jz(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_178
tff(fact_10391_ATP_Olambda__179,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] :
      aa_int_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_kc(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_179
tff(fact_10392_ATP_Olambda__180,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ha(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_180
tff(fact_10393_ATP_Olambda__181,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_he(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_181
tff(fact_10394_ATP_Olambda__182,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_gv(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_182
tff(fact_10395_ATP_Olambda__183,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gy(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_183
tff(fact_10396_ATP_Olambda__184,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hb(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_184
tff(fact_10397_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gw(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_185
tff(fact_10398_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hc(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_186
tff(fact_10399_ATP_Olambda__187,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_gx(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_187
tff(fact_10400_ATP_Olambda__188,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_gz(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_188
tff(fact_10401_ATP_Olambda__189,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_hd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_189
tff(fact_10402_ATP_Olambda__190,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_hk(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_190
tff(fact_10403_ATP_Olambda__191,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ho(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_191
tff(fact_10404_ATP_Olambda__192,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_hf(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_192
tff(fact_10405_ATP_Olambda__193,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_hi(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_193
tff(fact_10406_ATP_Olambda__194,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hl(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_194
tff(fact_10407_ATP_Olambda__195,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hg(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_195
tff(fact_10408_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_196
tff(fact_10409_ATP_Olambda__197,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_hh(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_197
tff(fact_10410_ATP_Olambda__198,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_hj(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_198
tff(fact_10411_ATP_Olambda__199,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_hn(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_199
tff(fact_10412_ATP_Olambda__200,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_pw(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_200
tff(fact_10413_ATP_Olambda__201,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_px(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_201
tff(fact_10414_ATP_Olambda__202,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_pv(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_202
tff(fact_10415_ATP_Olambda__203,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_jo(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_203
tff(fact_10416_ATP_Olambda__204,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_js(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_204
tff(fact_10417_ATP_Olambda__205,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_o,Uub: real] :
      aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_jj(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_205
tff(fact_10418_ATP_Olambda__206,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_jm(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_206
tff(fact_10419_ATP_Olambda__207,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_jp(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_207
tff(fact_10420_ATP_Olambda__208,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
      aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_jk(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_208
tff(fact_10421_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_jq(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_209
tff(fact_10422_ATP_Olambda__210,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_o,Uub: int] :
      aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_jl(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_210
tff(fact_10423_ATP_Olambda__211,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_jn(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_211
tff(fact_10424_ATP_Olambda__212,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_jr(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_212
tff(fact_10425_ATP_Olambda__213,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_te(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_213
tff(fact_10426_ATP_Olambda__214,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_sg(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_214
tff(fact_10427_ATP_Olambda__215,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_pu(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_215
tff(fact_10428_ATP_Olambda__216,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ro(Uu),Uua),Uub)
    <=> produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_rn(Uua),Uub),quotient_of(Uu)) ) ).

% ATP.lambda_216
tff(fact_10429_ATP_Olambda__217,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_rm(Uu),Uua),Uub)
    <=> produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_rl(Uua),Uub),quotient_of(Uu)) ) ).

% ATP.lambda_217
tff(fact_10430_ATP_Olambda__218,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ra(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qz(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_218
tff(fact_10431_ATP_Olambda__219,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qy(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qx(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_219
tff(fact_10432_ATP_Olambda__220,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qw(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qv(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_220
tff(fact_10433_ATP_Olambda__221,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qu(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qt(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_221
tff(fact_10434_ATP_Olambda__222,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ru(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_222
tff(fact_10435_ATP_Olambda__223,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_rv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_223
tff(fact_10436_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_my(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_224
tff(fact_10437_ATP_Olambda__225,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_na(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_225
tff(fact_10438_ATP_Olambda__226,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mz(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_226
tff(fact_10439_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sd(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_227
tff(fact_10440_ATP_Olambda__228,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_on(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_228
tff(fact_10441_ATP_Olambda__229,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_uf(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_229
tff(fact_10442_ATP_Olambda__230,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ug(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_230
tff(fact_10443_ATP_Olambda__231,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_dv(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_231
tff(fact_10444_ATP_Olambda__232,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_du(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_232
tff(fact_10445_ATP_Olambda__233,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_dw(Uu),Uua),Uub)
    <=> ( aa_set_o_o(ord_less_eq_set_o(set_o2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_233
tff(fact_10446_ATP_Olambda__234,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_dy(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_234
tff(fact_10447_ATP_Olambda__235,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_dx(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_235
tff(fact_10448_ATP_Olambda__236,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_dq(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_236
tff(fact_10449_ATP_Olambda__237,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_dp(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_237
tff(fact_10450_ATP_Olambda__238,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_dr(Uu),Uua),Uub)
    <=> ( aa_set_o_o(ord_less_eq_set_o(set_o2(Uub)),Uu)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_238
tff(fact_10451_ATP_Olambda__239,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_dt(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_239
tff(fact_10452_ATP_Olambda__240,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_ds(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_240
tff(fact_10453_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ue(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_241
tff(fact_10454_ATP_Olambda__242,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_pq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_242
tff(fact_10455_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_rh(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(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_243
tff(fact_10456_ATP_Olambda__244,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_pr(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_244
tff(fact_10457_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ox(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_245
tff(fact_10458_ATP_Olambda__246,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_cx(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_246
tff(fact_10459_ATP_Olambda__247,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_cy(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_247
tff(fact_10460_ATP_Olambda__248,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_cz(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_248
tff(fact_10461_ATP_Olambda__249,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_da(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_249
tff(fact_10462_ATP_Olambda__250,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_dc(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_250
tff(fact_10463_ATP_Olambda__251,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_db(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_251
tff(fact_10464_ATP_Olambda__252,axiom,
    ! [Uu: complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_252
tff(fact_10465_ATP_Olambda__253,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_253
tff(fact_10466_ATP_Olambda__254,axiom,
    ! [Uu: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_ce(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_254
tff(fact_10467_ATP_Olambda__255,axiom,
    ! [Uu: $o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_cb((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        | aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_255
tff(fact_10468_ATP_Olambda__256,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_cf(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_256
tff(fact_10469_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_257
tff(fact_10470_ATP_Olambda__258,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ch(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_258
tff(fact_10471_ATP_Olambda__259,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ag(Uu),Uua),Uub)
    <=> ( aa_nat_o(vEBT_vebt_member(Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_259
tff(fact_10472_ATP_Olambda__260,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_nat_o(vEBT_vebt_member(Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_260
tff(fact_10473_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_tn(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_261
tff(fact_10474_ATP_Olambda__262,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_to(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uua)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uu) ) ) ).

% ATP.lambda_262
tff(fact_10475_ATP_Olambda__263,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_dd(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_263
tff(fact_10476_ATP_Olambda__264,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_de(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_264
tff(fact_10477_ATP_Olambda__265,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_df(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_265
tff(fact_10478_ATP_Olambda__266,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_dg(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_266
tff(fact_10479_ATP_Olambda__267,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_di(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_267
tff(fact_10480_ATP_Olambda__268,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_dh(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_268
tff(fact_10481_ATP_Olambda__269,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ax(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_269
tff(fact_10482_ATP_Olambda__270,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ay(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_270
tff(fact_10483_ATP_Olambda__271,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_az(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_271
tff(fact_10484_ATP_Olambda__272,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ba(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_272
tff(fact_10485_ATP_Olambda__273,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bc(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_273
tff(fact_10486_ATP_Olambda__274,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bb(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_274
tff(fact_10487_ATP_Olambda__275,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_br(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_275
tff(fact_10488_ATP_Olambda__276,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_276
tff(fact_10489_ATP_Olambda__277,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bs((Uu)),Uua),(Uub))
    <=> ( ( (Uu)
        <=> (Uub) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_277
tff(fact_10490_ATP_Olambda__278,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_278
tff(fact_10491_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_279
tff(fact_10492_ATP_Olambda__280,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_280
tff(fact_10493_ATP_Olambda__281,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bl(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_281
tff(fact_10494_ATP_Olambda__282,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_bj(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_282
tff(fact_10495_ATP_Olambda__283,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bm((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_283
tff(fact_10496_ATP_Olambda__284,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bk(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_284
tff(fact_10497_ATP_Olambda__285,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_285
tff(fact_10498_ATP_Olambda__286,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_286
tff(fact_10499_ATP_Olambda__287,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_ff(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_287
tff(fact_10500_ATP_Olambda__288,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_fn(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_288
tff(fact_10501_ATP_Olambda__289,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_ez(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_289
tff(fact_10502_ATP_Olambda__290,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_fh(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_290
tff(fact_10503_ATP_Olambda__291,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_fp(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_291
tff(fact_10504_ATP_Olambda__292,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_fb(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_292
tff(fact_10505_ATP_Olambda__293,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_fj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_293
tff(fact_10506_ATP_Olambda__294,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_fr(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_294
tff(fact_10507_ATP_Olambda__295,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_fd(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_295
tff(fact_10508_ATP_Olambda__296,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_fl(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_296
tff(fact_10509_ATP_Olambda__297,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_el(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_297
tff(fact_10510_ATP_Olambda__298,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_et(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_298
tff(fact_10511_ATP_Olambda__299,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_ef(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_299
tff(fact_10512_ATP_Olambda__300,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_en(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_300
tff(fact_10513_ATP_Olambda__301,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_ev(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_301
tff(fact_10514_ATP_Olambda__302,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_eh(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_302
tff(fact_10515_ATP_Olambda__303,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_ep(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_303
tff(fact_10516_ATP_Olambda__304,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_ex(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_304
tff(fact_10517_ATP_Olambda__305,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_ej(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_305
tff(fact_10518_ATP_Olambda__306,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_er(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_306
tff(fact_10519_ATP_Olambda__307,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_an(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ~ aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_307
tff(fact_10520_ATP_Olambda__308,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_ao(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_308
tff(fact_10521_ATP_Olambda__309,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_ap(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_309
tff(fact_10522_ATP_Olambda__310,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & ~ aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_310
tff(fact_10523_ATP_Olambda__311,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_311
tff(fact_10524_ATP_Olambda__312,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_312
tff(fact_10525_ATP_Olambda__313,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_tm(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_313
tff(fact_10526_ATP_Olambda__314,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_ts(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu) ) ).

% ATP.lambda_314
tff(fact_10527_ATP_Olambda__315,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_tq(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu) ) ).

% ATP.lambda_315
tff(fact_10528_ATP_Olambda__316,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_ee(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_316
tff(fact_10529_ATP_Olambda__317,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_pk(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_317
tff(fact_10530_ATP_Olambda__318,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_sv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_318
tff(fact_10531_ATP_Olambda__319,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_se(Uu),Uua),Uub) = aa_real_real(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_319
tff(fact_10532_ATP_Olambda__320,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ok(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_320
tff(fact_10533_ATP_Olambda__321,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pa(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_321
tff(fact_10534_ATP_Olambda__322,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_do(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_322
tff(fact_10535_ATP_Olambda__323,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ti(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_323
tff(fact_10536_ATP_Olambda__324,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_324
tff(fact_10537_ATP_Olambda__325,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_325
tff(fact_10538_ATP_Olambda__326,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_ml(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_326
tff(fact_10539_ATP_Olambda__327,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_mp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_327
tff(fact_10540_ATP_Olambda__328,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ms(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_328
tff(fact_10541_ATP_Olambda__329,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_mm(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_329
tff(fact_10542_ATP_Olambda__330,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_330
tff(fact_10543_ATP_Olambda__331,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_mt(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_331
tff(fact_10544_ATP_Olambda__332,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_mn(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_332
tff(fact_10545_ATP_Olambda__333,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mu(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_333
tff(fact_10546_ATP_Olambda__334,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_mo(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_334
tff(fact_10547_ATP_Olambda__335,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_mr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_335
tff(fact_10548_ATP_Olambda__336,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_336
tff(fact_10549_ATP_Olambda__337,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_jc(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_337
tff(fact_10550_ATP_Olambda__338,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_je(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_338
tff(fact_10551_ATP_Olambda__339,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_339
tff(fact_10552_ATP_Olambda__340,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_jb(Uu),Uua),Uub) = aa_int_int(plus_plus_int(aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_340
tff(fact_10553_ATP_Olambda__341,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_rt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_341
tff(fact_10554_ATP_Olambda__342,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ft(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
       => aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_342
tff(fact_10555_ATP_Olambda__343,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_fu(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_343
tff(fact_10556_ATP_Olambda__344,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_fv(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_344
tff(fact_10557_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_fx(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_345
tff(fact_10558_ATP_Olambda__346,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_fw(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_346
tff(fact_10559_ATP_Olambda__347,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        | aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_347
tff(fact_10560_ATP_Olambda__348,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ct(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_348
tff(fact_10561_ATP_Olambda__349,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_cu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_349
tff(fact_10562_ATP_Olambda__350,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cw(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_350
tff(fact_10563_ATP_Olambda__351,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cv(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_351
tff(fact_10564_ATP_Olambda__352,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_dj(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_352
tff(fact_10565_ATP_Olambda__353,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_dk(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_353
tff(fact_10566_ATP_Olambda__354,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_dl(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_354
tff(fact_10567_ATP_Olambda__355,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_dn(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_355
tff(fact_10568_ATP_Olambda__356,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_dm(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_356
tff(fact_10569_ATP_Olambda__357,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] : aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_id(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uu,Uub)),aa_o_real(zero_n3304061248610475627l_real,aa_complex_o(Uua,Uub))) ).

% ATP.lambda_357
tff(fact_10570_ATP_Olambda__358,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_o,Uub: complex] : aa_complex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_ig(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_complex_o(Uua,Uub))) ).

% ATP.lambda_358
tff(fact_10571_ATP_Olambda__359,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_o,Uub: complex] : aa_complex_nat(aa_fun7518244829075205762ex_nat(aTP_Lamm_ik(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_complex_nat(Uu,Uub)),aa_o_nat(zero_n2687167440665602831ol_nat,aa_complex_o(Uua,Uub))) ).

% ATP.lambda_359
tff(fact_10572_ATP_Olambda__360,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] : aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_ib(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uu,Uub)),aa_o_real(zero_n3304061248610475627l_real,aa_real_o(Uua,Uub))) ).

% ATP.lambda_360
tff(fact_10573_ATP_Olambda__361,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] : aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_ie(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_real_o(Uua,Uub))) ).

% ATP.lambda_361
tff(fact_10574_ATP_Olambda__362,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_o,Uub: real] : aa_real_nat(aa_fun5513133441921015042al_nat(aTP_Lamm_ii(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_real_nat(Uu,Uub)),aa_o_nat(zero_n2687167440665602831ol_nat,aa_real_o(Uua,Uub))) ).

% ATP.lambda_362
tff(fact_10575_ATP_Olambda__363,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] : aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ih(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(Uua,Uub))) ).

% ATP.lambda_363
tff(fact_10576_ATP_Olambda__364,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] : aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_ic(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uu,Uub)),aa_o_real(zero_n3304061248610475627l_real,aa_int_o(Uua,Uub))) ).

% ATP.lambda_364
tff(fact_10577_ATP_Olambda__365,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] : aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_if(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_int_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_int_o(Uua,Uub))) ).

% ATP.lambda_365
tff(fact_10578_ATP_Olambda__366,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] : aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_ij(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uu,Uub)),aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uua,Uub))) ).

% ATP.lambda_366
tff(fact_10579_ATP_Olambda__367,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_367
tff(fact_10580_ATP_Olambda__368,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jh(Uu),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_368
tff(fact_10581_ATP_Olambda__369,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: code_integer,Uub: complex] : aa_com4217951221296830281nteger(aa_Cod1660996421839551523nteger(aTP_Lamm_ld(Uu),Uua),Uub) = divide6298287555418463151nteger(aa_com4217951221296830281nteger(Uu,Uub),Uua) ).

% ATP.lambda_369
tff(fact_10582_ATP_Olambda__370,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_jf(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_370
tff(fact_10583_ATP_Olambda__371,axiom,
    ! [Uu: fun_complex_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_lp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_complex_nat(Uu,Uub)),Uua) ).

% ATP.lambda_371
tff(fact_10584_ATP_Olambda__372,axiom,
    ! [Uu: fun_complex_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_lv(Uu),Uua),Uub) = aa_int_int(divide_divide_int(aa_complex_int(Uu,Uub)),Uua) ).

% ATP.lambda_372
tff(fact_10585_ATP_Olambda__373,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: code_integer,Uub: real] : aa_real_Code_integer(aa_Cod7053582236991556129nteger(aTP_Lamm_kx(Uu),Uua),Uub) = divide6298287555418463151nteger(aa_real_Code_integer(Uu,Uub),Uua) ).

% ATP.lambda_373
tff(fact_10586_ATP_Olambda__374,axiom,
    ! [Uu: fun_real_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_lj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_real_nat(Uu,Uub)),Uua) ).

% ATP.lambda_374
tff(fact_10587_ATP_Olambda__375,axiom,
    ! [Uu: fun_real_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_ls(Uu),Uua),Uub) = aa_int_int(divide_divide_int(aa_real_int(Uu,Uub)),Uua) ).

% ATP.lambda_375
tff(fact_10588_ATP_Olambda__376,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_lg(Uu),Uua),Uub) = divide6298287555418463151nteger(aa_nat_Code_integer(Uu,Uub),Uua) ).

% ATP.lambda_376
tff(fact_10589_ATP_Olambda__377,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_377
tff(fact_10590_ATP_Olambda__378,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ly(Uu),Uua),Uub) = aa_int_int(divide_divide_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_378
tff(fact_10591_ATP_Olambda__379,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] : aa_int_Code_integer(aa_Cod1949805079358391713nteger(aTP_Lamm_la(Uu),Uua),Uub) = divide6298287555418463151nteger(aa_int_Code_integer(Uu,Uub),Uua) ).

% ATP.lambda_379
tff(fact_10592_ATP_Olambda__380,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_lm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(Uu,Uub)),Uua) ).

% ATP.lambda_380
tff(fact_10593_ATP_Olambda__381,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_iu(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_381
tff(fact_10594_ATP_Olambda__382,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_382
tff(fact_10595_ATP_Olambda__383,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_iv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_383
tff(fact_10596_ATP_Olambda__384,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_it(Uu),Uua),Uub) = aa_int_int(times_times_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_384
tff(fact_10597_ATP_Olambda__385,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rq(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_385
tff(fact_10598_ATP_Olambda__386,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tb(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_386
tff(fact_10599_ATP_Olambda__387,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_rs(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_387
tff(fact_10600_ATP_Olambda__388,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bw(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_388
tff(fact_10601_ATP_Olambda__389,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_389
tff(fact_10602_ATP_Olambda__390,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bv((Uu)),Uua),(Uub))
    <=> ( ~ ( (Uub)
          <=> (Uu) )
       => aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_390
tff(fact_10603_ATP_Olambda__391,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_by(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_391
tff(fact_10604_ATP_Olambda__392,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bz(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_392
tff(fact_10605_ATP_Olambda__393,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ca(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_393
tff(fact_10606_ATP_Olambda__394,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ht(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_o_real(zero_n3304061248610475627l_real,aa_complex_o(Uu,Uub))),aa_complex_real(Uua,Uub)) ).

% ATP.lambda_394
tff(fact_10607_ATP_Olambda__395,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_hw(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_o_rat(zero_n2052037380579107095ol_rat,aa_complex_o(Uu,Uub))),aa_complex_rat(Uua,Uub)) ).

% ATP.lambda_395
tff(fact_10608_ATP_Olambda__396,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_ia(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_complex_o(Uu,Uub))),aa_complex_nat(Uua,Uub)) ).

% ATP.lambda_396
tff(fact_10609_ATP_Olambda__397,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_hr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_o_real(zero_n3304061248610475627l_real,aa_real_o(Uu,Uub))),aa_real_real(Uua,Uub)) ).

% ATP.lambda_397
tff(fact_10610_ATP_Olambda__398,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_hu(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_o_rat(zero_n2052037380579107095ol_rat,aa_real_o(Uu,Uub))),aa_real_rat(Uua,Uub)) ).

% ATP.lambda_398
tff(fact_10611_ATP_Olambda__399,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_hy(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_real_o(Uu,Uub))),aa_real_nat(Uua,Uub)) ).

% ATP.lambda_399
tff(fact_10612_ATP_Olambda__400,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hx(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(Uu,Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_400
tff(fact_10613_ATP_Olambda__401,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_hs(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_o_real(zero_n3304061248610475627l_real,aa_int_o(Uu,Uub))),aa_int_real(Uua,Uub)) ).

% ATP.lambda_401
tff(fact_10614_ATP_Olambda__402,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_hv(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_o_rat(zero_n2052037380579107095ol_rat,aa_int_o(Uu,Uub))),aa_int_rat(Uua,Uub)) ).

% ATP.lambda_402
tff(fact_10615_ATP_Olambda__403,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_hz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uu,Uub))),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_403
tff(fact_10616_ATP_Olambda__404,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_404
tff(fact_10617_ATP_Olambda__405,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_nf(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_405
tff(fact_10618_ATP_Olambda__406,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ni(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_406
tff(fact_10619_ATP_Olambda__407,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ng(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_407
tff(fact_10620_ATP_Olambda__408,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nh(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_408
tff(fact_10621_ATP_Olambda__409,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(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_409
tff(fact_10622_ATP_Olambda__410,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_iy(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_410
tff(fact_10623_ATP_Olambda__411,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ja(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_411
tff(fact_10624_ATP_Olambda__412,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_iz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_412
tff(fact_10625_ATP_Olambda__413,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ix(Uu),Uua),Uub) = aa_int_int(times_times_int(Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_413
tff(fact_10626_ATP_Olambda__414,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: code_integer,Uub: complex] :
      ( aa_complex_o(aa_Cod7908732550325689320plex_o(aTP_Lamm_le(Uu),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_com4217951221296830281nteger(Uu,Uub)) ) ).

% ATP.lambda_414
tff(fact_10627_ATP_Olambda__415,axiom,
    ! [Uu: fun_complex_nat,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_lq(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_complex_nat(Uu,Uub)) ) ).

% ATP.lambda_415
tff(fact_10628_ATP_Olambda__416,axiom,
    ! [Uu: fun_complex_int,Uua: int,Uub: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aTP_Lamm_lw(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_complex_int(Uu,Uub)) ) ).

% ATP.lambda_416
tff(fact_10629_ATP_Olambda__417,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: code_integer,Uub: real] :
      ( aa_real_o(aa_Cod1315720040876898922real_o(aTP_Lamm_ky(Uu),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_real_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_417
tff(fact_10630_ATP_Olambda__418,axiom,
    ! [Uu: fun_real_nat,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_lk(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_real_nat(Uu,Uub)) ) ).

% ATP.lambda_418
tff(fact_10631_ATP_Olambda__419,axiom,
    ! [Uu: fun_real_int,Uua: int,Uub: real] :
      ( aa_real_o(aa_int_fun_real_o(aTP_Lamm_lt(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_real_int(Uu,Uub)) ) ).

% ATP.lambda_419
tff(fact_10632_ATP_Olambda__420,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aTP_Lamm_lh(Uu),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_nat_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_420
tff(fact_10633_ATP_Olambda__421,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_lz(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_421
tff(fact_10634_ATP_Olambda__422,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aTP_Lamm_lb(Uu),Uua),Uub)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_int_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_422
tff(fact_10635_ATP_Olambda__423,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_ln(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_423
tff(fact_10636_ATP_Olambda__424,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_tg(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua)) ) ).

% ATP.lambda_424
tff(fact_10637_ATP_Olambda__425,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ne(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_425
tff(fact_10638_ATP_Olambda__426,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_tk(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_426
tff(fact_10639_ATP_Olambda__427,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nd(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_427
tff(fact_10640_ATP_Olambda__428,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_ua(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_428
tff(fact_10641_ATP_Olambda__429,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_su(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_429
tff(fact_10642_ATP_Olambda__430,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: code_integer,Uub: complex] :
      ( aa_complex_o(aa_Cod7908732550325689320plex_o(aTP_Lamm_lf(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_com4217951221296830281nteger(Uu,Uub)) ) ).

% ATP.lambda_430
tff(fact_10643_ATP_Olambda__431,axiom,
    ! [Uu: fun_complex_nat,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_lr(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_complex_nat(Uu,Uub)) ) ).

% ATP.lambda_431
tff(fact_10644_ATP_Olambda__432,axiom,
    ! [Uu: fun_complex_int,Uua: int,Uub: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aTP_Lamm_lx(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_complex_int(Uu,Uub)) ) ).

% ATP.lambda_432
tff(fact_10645_ATP_Olambda__433,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: code_integer,Uub: real] :
      ( aa_real_o(aa_Cod1315720040876898922real_o(aTP_Lamm_kz(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_real_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_433
tff(fact_10646_ATP_Olambda__434,axiom,
    ! [Uu: fun_real_nat,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_ll(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_real_nat(Uu,Uub)) ) ).

% ATP.lambda_434
tff(fact_10647_ATP_Olambda__435,axiom,
    ! [Uu: fun_real_int,Uua: int,Uub: real] :
      ( aa_real_o(aa_int_fun_real_o(aTP_Lamm_lu(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_real_int(Uu,Uub)) ) ).

% ATP.lambda_435
tff(fact_10648_ATP_Olambda__436,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aTP_Lamm_li(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_nat_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_436
tff(fact_10649_ATP_Olambda__437,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_ma(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_437
tff(fact_10650_ATP_Olambda__438,axiom,
    ! [Uu: fun_int_Code_integer,Uua: code_integer,Uub: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aTP_Lamm_lc(Uu),Uua),Uub)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),aa_int_Code_integer(Uu,Uub)) ) ).

% ATP.lambda_438
tff(fact_10651_ATP_Olambda__439,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_lo(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_439
tff(fact_10652_ATP_Olambda__440,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) = $ite(Uuc = Uu,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_440
tff(fact_10653_ATP_Olambda__441,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_mc(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_441
tff(fact_10654_ATP_Olambda__442,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_md(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_442
tff(fact_10655_ATP_Olambda__443,axiom,
    ! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] :
      aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_me(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_int(Uua,Uuc),aa_complex_int(Uub,Uuc)) ).

% ATP.lambda_443
tff(fact_10656_ATP_Olambda__444,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_mf(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_real(Uua,Uuc),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_444
tff(fact_10657_ATP_Olambda__445,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_mg(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_rat(Uua,Uuc),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_445
tff(fact_10658_ATP_Olambda__446,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_mh(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_nat(Uua,Uuc),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_446
tff(fact_10659_ATP_Olambda__447,axiom,
    ! [Uu: real,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] :
      aa_real_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_mi(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_int(Uua,Uuc),aa_real_int(Uub,Uuc)) ).

% ATP.lambda_447
tff(fact_10660_ATP_Olambda__448,axiom,
    ! [Uu: $o,Uua: fun_o_real,Uub: fun_o_real,Uuc: $o] :
      aa_o_real(aa_fun688608109223240316o_real(aa_fun1133276423545935902o_real(aTP_Lamm_mj((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_real(Uua,(Uuc)),aa_o_real(Uub,(Uuc))) ).

% ATP.lambda_448
tff(fact_10661_ATP_Olambda__449,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_mk((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_rat(Uua,(Uuc)),aa_o_rat(Uub,(Uuc))) ).

% ATP.lambda_449
tff(fact_10662_ATP_Olambda__450,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kp(Uu),Uua),Uub),Uuc) = $ite(aa_complex_o(Uu,Uuc),aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_450
tff(fact_10663_ATP_Olambda__451,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_ks(Uu),Uua),Uub),Uuc) = $ite(aa_complex_o(Uu,Uuc),aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_451
tff(fact_10664_ATP_Olambda__452,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_kv(Uu),Uua),Uub),Uuc) = $ite(aa_complex_o(Uu,Uuc),aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_452
tff(fact_10665_ATP_Olambda__453,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_kn(Uu),Uua),Uub),Uuc) = $ite(aa_real_o(Uu,Uuc),aa_real_real(Uua,Uuc),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_453
tff(fact_10666_ATP_Olambda__454,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_kq(Uu),Uua),Uub),Uuc) = $ite(aa_real_o(Uu,Uuc),aa_real_rat(Uua,Uuc),aa_real_rat(Uub,Uuc)) ).

% ATP.lambda_454
tff(fact_10667_ATP_Olambda__455,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_kt(Uu),Uua),Uub),Uuc) = $ite(aa_real_o(Uu,Uuc),aa_real_nat(Uua,Uuc),aa_real_nat(Uub,Uuc)) ).

% ATP.lambda_455
tff(fact_10668_ATP_Olambda__456,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_int,Uub: fun_real_int,Uuc: real] :
      aa_real_int(aa_fun8270430325945277160al_int(aa_fun7818883909550569274al_int(aTP_Lamm_kw(Uu),Uua),Uub),Uuc) = $ite(aa_real_o(Uu,Uuc),aa_real_int(Uua,Uuc),aa_real_int(Uub,Uuc)) ).

% ATP.lambda_456
tff(fact_10669_ATP_Olambda__457,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_ko(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_real(Uua,Uuc),aa_int_real(Uub,Uuc)) ).

% ATP.lambda_457
tff(fact_10670_ATP_Olambda__458,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_kr(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_458
tff(fact_10671_ATP_Olambda__459,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ku(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_nat(Uua,Uuc),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_459
tff(fact_10672_ATP_Olambda__460,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_is(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ir(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_460
tff(fact_10673_ATP_Olambda__461,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_iq(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ip(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_461
tff(fact_10674_ATP_Olambda__462,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_io(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_in(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_462
tff(fact_10675_ATP_Olambda__463,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_im(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_il(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_463
tff(fact_10676_ATP_Olambda__464,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_pc(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_pb(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_464
tff(fact_10677_ATP_Olambda__465,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sa(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_465
tff(fact_10678_ATP_Olambda__466,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_466
tff(fact_10679_ATP_Olambda__467,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rw(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_467
tff(fact_10680_ATP_Olambda__468,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_468
tff(fact_10681_ATP_Olambda__469,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qa(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_469
tff(fact_10682_ATP_Olambda__470,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oz(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_470
tff(fact_10683_ATP_Olambda__471,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oy(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_471
tff(fact_10684_ATP_Olambda__472,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_rl(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uua),Uub)) ) ).

% ATP.lambda_472
tff(fact_10685_ATP_Olambda__473,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_qe(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_473
tff(fact_10686_ATP_Olambda__474,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_rn(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uua),Uub)) ) ).

% ATP.lambda_474
tff(fact_10687_ATP_Olambda__475,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_qc(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_475
tff(fact_10688_ATP_Olambda__476,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qg(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_476
tff(fact_10689_ATP_Olambda__477,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qi(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_477
tff(fact_10690_ATP_Olambda__478,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_em(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_478
tff(fact_10691_ATP_Olambda__479,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_eu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_479
tff(fact_10692_ATP_Olambda__480,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_eg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(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_480
tff(fact_10693_ATP_Olambda__481,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_eo(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(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_481
tff(fact_10694_ATP_Olambda__482,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_ew(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_482
tff(fact_10695_ATP_Olambda__483,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_ei(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_483
tff(fact_10696_ATP_Olambda__484,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_eq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_484
tff(fact_10697_ATP_Olambda__485,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ey(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_485
tff(fact_10698_ATP_Olambda__486,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ek(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_486
tff(fact_10699_ATP_Olambda__487,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_es(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_487
tff(fact_10700_ATP_Olambda__488,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_fg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(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_488
tff(fact_10701_ATP_Olambda__489,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_fo(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_489
tff(fact_10702_ATP_Olambda__490,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_fa(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(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_490
tff(fact_10703_ATP_Olambda__491,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_fi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_491
tff(fact_10704_ATP_Olambda__492,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_fq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_492
tff(fact_10705_ATP_Olambda__493,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_fc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(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_493
tff(fact_10706_ATP_Olambda__494,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_fk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_494
tff(fact_10707_ATP_Olambda__495,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_fs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_495
tff(fact_10708_ATP_Olambda__496,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_fe(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(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_496
tff(fact_10709_ATP_Olambda__497,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_fm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_497
tff(fact_10710_ATP_Olambda__498,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_pb(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_498
tff(fact_10711_ATP_Olambda__499,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_in(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_499
tff(fact_10712_ATP_Olambda__500,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_ir(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_500
tff(fact_10713_ATP_Olambda__501,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_ip(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_501
tff(fact_10714_ATP_Olambda__502,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_il(Uu),Uua),Uub),Uuc) = aa_int_int(times_times_int(aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_502
tff(fact_10715_ATP_Olambda__503,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_nk(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_503
tff(fact_10716_ATP_Olambda__504,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_nj(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_504
tff(fact_10717_ATP_Olambda__505,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qx(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(minus_minus_int(aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uub),Uua)),aa_int_int(times_times_int(Uua),Uuc))) ).

% ATP.lambda_505
tff(fact_10718_ATP_Olambda__506,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qz(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uub),Uua)),aa_int_int(times_times_int(Uua),Uuc))) ).

% ATP.lambda_506
tff(fact_10719_ATP_Olambda__507,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qv(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(times_times_int(Uu),Uub),aa_int_int(times_times_int(Uua),Uuc))) ).

% ATP.lambda_507
tff(fact_10720_ATP_Olambda__508,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qt(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(times_times_int(Uu),Uuc),aa_int_int(times_times_int(Uua),Uub))) ).

% ATP.lambda_508
tff(fact_10721_ATP_Olambda__509,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_rz(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ry(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_509
tff(fact_10722_ATP_Olambda__510,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_tp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I)),X_1)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I) ) )
        & $ite(
            Uue = Uuf,
            ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Uua))
             => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X4)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X4)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_510
tff(fact_10723_ATP_Olambda__511,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cn,Uu) = Uu ).

% ATP.lambda_511
tff(fact_10724_ATP_Olambda__512,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_co,Uu) = Uu ).

% ATP.lambda_512
tff(fact_10725_ATP_Olambda__513,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cp,Uu) = Uu ).

% ATP.lambda_513
tff(fact_10726_ATP_Olambda__514,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cq,Uu) = Uu ).

% ATP.lambda_514
tff(fact_10727_ATP_Olambda__515,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cr,Uu) = Uu ).

% ATP.lambda_515
tff(fact_10728_ATP_Olambda__516,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ci,Uu) = zero_zero_complex ).

% ATP.lambda_516
tff(fact_10729_ATP_Olambda__517,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_cj,Uu) = zero_zero_real ).

% ATP.lambda_517
tff(fact_10730_ATP_Olambda__518,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ck,Uu) = zero_zero_rat ).

% ATP.lambda_518
tff(fact_10731_ATP_Olambda__519,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_gu,Uu) = zero_zero_real ).

% ATP.lambda_519
tff(fact_10732_ATP_Olambda__520,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cl,Uu) = zero_zero_nat ).

% ATP.lambda_520
tff(fact_10733_ATP_Olambda__521,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cm,Uu) = zero_zero_int ).

% ATP.lambda_521
tff(fact_10734_ATP_Olambda__522,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_bf,Uu)
    <=> $false ) ).

% ATP.lambda_522
tff(fact_10735_ATP_Olambda__523,axiom,
    ! [Uu: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_bd,Uu)
    <=> $false ) ).

% ATP.lambda_523
tff(fact_10736_ATP_Olambda__524,axiom,
    ! [Uu: $o] :
      ( aa_o_o(aTP_Lamm_bg,(Uu))
    <=> $false ) ).

% ATP.lambda_524
tff(fact_10737_ATP_Olambda__525,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_be,Uu)
    <=> $false ) ).

% ATP.lambda_525
tff(fact_10738_ATP_Olambda__526,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_bh,Uu)
    <=> $false ) ).

% ATP.lambda_526
tff(fact_10739_ATP_Olambda__527,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_bi,Uu)
    <=> $false ) ).

% ATP.lambda_527
tff(fact_10740_ATP_Olambda__528,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_pp,Uu)
    <=> $true ) ).

% ATP.lambda_528

% Helper facts (13)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X10: $o,Y: $o] :
      ( ( (X10) != (Y) )
      | aa_o_o(fequal_o((X10)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X10: $o,Y: $o] :
      ( ~ aa_o_o(fequal_o((X10)),(Y))
      | ( (X10) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X10: int,Y: int] :
      ( ( X10 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X10),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X10: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X10),Y)
      | ( X10 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X10: nat,Y: nat] :
      ( ( X10 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X10),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X10: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X10),Y)
      | ( X10 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X10: real,Y: real] :
      ( ( X10 != Y )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X10),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X10: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X10),Y)
      | ( X10 = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X9: real] : aa_real_o(P,X9) ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X10: set_nat,Y: set_nat] :
      ( ( X10 != Y )
      | aa_set_nat_o(fequal_set_nat(X10),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X10: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(fequal_set_nat(X10),Y)
      | ( X10 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X10: list_nat,Y: list_nat] :
      ( ( X10 != Y )
      | aa_list_nat_o(fequal_list_nat(X10),Y) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X10: list_nat,Y: list_nat] :
      ( ~ aa_list_nat_o(fequal_list_nat(X10),Y)
      | ( X10 = Y ) ) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,i),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)) ).

tff(conj_1,conjecture,
    ( aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList2,i)),x)
  <=> aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,i)),x) ) ).

%------------------------------------------------------------------------------